Results on the growth of solutions of complex linear differential equations with meromorphic coefficients

1. Introduction and main results

Throughout this work, we assume that the reader knows the standard notations and the fundamental results of the Nevanlinna value distribution theory of meromorphic functions as the order and the hyper-order of growth, the convergence exponents of the zero-sequence and of distinct zeros, the hyper convergence exponents of the zero-sequence and the distinct zeros of a meromorphic function \(f,\) see [2-5].

We recall also the following definitions. The linear measure of a set \(E\subset \left[ 0,+\infty \right)\) is defined as \(m\left( E\right) =\int_{0}^{+\infty }\chi _{E}\left( t\right) dt\) and the logarithmic measure of a set \(F\subset \left[ 1,+\infty \right)\) is defined by \(lm\left( F\right) =\int_{1}^{+\infty }\frac{\chi _{F}\left( t\right) }{t}% dt,\) where \(\chi _{H}\left( t\right)\) is the characteristic function of a set \(H\).

For results on the growth of solutions of the complex linear differential equation \[f^{\left( k\right) }+A_{k-1}\left( z\right) e^{P_{k-1}\left( z\right) }f^{\left( k-1\right) }+\cdots +A_{1}\left( z\right) e^{P_{1}\left( z\right) }f^{\prime }+A_{0}\left( z\right) e^{P_{0}\left( z\right) }f=0,\] where \(P_{j}\left( z\right) =a_{j,n}z^{n}+\cdots +a_{j,0}\) are polynomials with degree \(n\geq 1\), \(a_{j,q}\) \((j=0,1,…,k-1;q=0,1,…,n)\) are complex numbers and \(A_{j}(z)\) \((\not\equiv 0)\), \(\mathit{\ }(j=0,1,…,k-1)\) are entire or meromorphic functions of finite order less than \(n,\) the reader is referred to [1,6-8].

Recently, Hamani and Belaı̈di [1] studied the order of transcendental meromorphic solutions of the homogeneous and the non-homogeneous linear differential equations \[\ f^{\left( k\right) }+h_{k-1}\left( z\right) e^{P_{k-1}\left( z\right) }f^{\left( k-1\right) }+\cdots +h_{1}\left( z\right) e^{P_{1}\left( z\right) }f^{\prime }+h_{0}\left( z\right) e^{P_{0}\left( z\right) }f=0, \label{1.1}, \tag{1}\] \[\ f^{\left( k\right) }+h_{k-1}\left( z\right) e^{P_{k-1}\left( z\right) }f^{\left( k-1\right) }+\cdots +h_{1}\left( z\right) e^{P_{1}\left( z\right) }f^{\prime }+h_{0}\left( z\right) e^{P_{0}\left( z\right) }f=F, \label{1.2}, \tag{2}\] and have proved the following results;

In this paper, we continue to study the oscillation problem of solutions, we improve and extend Theorem 2 and Theorem 2 for equations of the form \[\ f^{\left( k\right) }+\left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left( z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right) f^{\left( k-1\right) } +\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right) }+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=0, \label{1.3}, \tag{3}\] and \[\ f^{\left( k\right) }+\left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left( z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right) f^{\left( k-1\right) } +\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right) }+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=F. \label{1.4}, \tag{4}\] We obtain the following results;

For the case of non-homogeneous equation, we have the following result;

2. Auxiliary lemmas for the proofs of the theorems

To prove our theorems, we need the following lemmas;

3. Proof of Theorem 3

First we prove that every transcendental meromorphic solution \(f\) of Eq. (3) is of order \(\rho \left( f\right) \geq n\) . Assume that \(f\) is a transcendental meromorphic solution of Eq. (3) with \(\rho \left( f\right) <n.\) We can rewrite Eq. (3) in the form \begin{equation} \left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left( z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right) f^{\left( k-1\right) } +\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right) }+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=-f^{\left( k\right) }. \tag{8}\end{equation}

Since \[\max \left\{ \rho \left( A_{j,i}\right) :j=0,1,…,k-1;i=1,2\right\} <n\] and \(\rho \left( f\right) <n,\) then \(A_{j,i}f^{\left( j\right) }\) \(\left( j=0,1,…,k-1;i=1,2\right)\) and \(f^{\left( k\right) }\) are meromorphic functions of finite order with \(\rho \left( A_{j,i}f^{\left( j\right) }\right)\) \(<n\) and \(\rho \left( f^{\left( k\right) }\right) <n.\) We have \(% A_{s,i}f^{\left( s\right) }\not\equiv 0\left( i=1,2\right)\). Indeed, if \(% A_{s,i}f^{\left( s\right) }\equiv 0\), it follows that \(f^{\left( s\right) }\equiv 0\). Then \(f\) has to be a polynomial of degree less than \(s\). This is a contradiction. Since \(a_{s,n}\neq b_{s,n}\) and \(% a_{j,n}=c_{j}a_{s,n},b_{j,n}=c_{j}^{\prime }b_{s,n},(0<c_{j}<1),(0<c_{j}^{\prime }<1),(j\neq s),a_{j,n}=c_{j}a_{s,n}\)  or \(a_{j,n}=c_{j}a_{d,n}(0<c_{j}<1),\) then \(a_{j,n}\neq b_{j,n}\), \(a_{s,n}\neq a_{j,n}\), \(b_{s,n}\neq b_{j,n}\) and therefore \(\deg \left( P_{s}-P_{j}\right) =\deg \left( Q_{s}-Q_{j}\right) =n.\) Thus, by (8) and Lemma 1, we find that \(\rho \left( -f^{\left( k\right) }\right) =n,\) this contradicts the fact \(\rho \left( f^{\left( k\right) }\right) <n.\) Consequently, every meromorphic solution \(f\) of Eq. (3) is transcendental with order \(\rho \left( f\right) \geq n\).

Assume that \(f\) is a transcendental meromorphic solution whose poles are of uniformly bounded multiplicities of Eq. (3). By Lemma 2, there exist a set \(E_{1}\) \(\subset (1,+\infty )\)  having finite logarithmic measure and a constant \(B>0\), such that for all \(% z\) satisfying \(|z|=r\notin \lbrack 0,1]\diagdown E_{1}\), we have \[\begin{equation} \left\vert \frac{f^{\left( j\right) }\left( z\right) }{f\left( z\right) }% \right\vert \leq B\left( T\left( 2r,f\right) \right) ^{k+1},\text{ }% j=1,2,…,k,\text{ }j\neq s. \end{equation}\tag{9}\]

By (3), it follows that the poles of \(f\) can only occur at the poles of \(A_{j,i}(z)\) \(\left( j=0,1,…,k-1;i=1,2\right)\). Note that the poles of \(f\) are of uniformly bounded multiplicities. Hence \[\lambda \left( \frac{1}{f}\right) \leq \max \left\{ \rho \left( A_{j,i}\right) :j=0,1,…,k-1;i=1,2\right\} <n.\] By Hadamard factorization theorem, we know that \(f\) can be written as \(f(z)=% \frac{g(z)}{d(z)},\) where \(g(z)\) and \(d(z)\) are entire functions with \[\lambda \left( d\right) =\rho \left( d\right) =\lambda \left( \frac{1}{f}% \right) <n\leq \rho \left( f\right) =\rho \left( g\right) .\] For each sufficiently large \(|z|=r\), let \(z_{r}=re^{i\theta _{r}}\) be a point satisfying \(\left\vert g(z_{r})\right\vert\) \(=M(r,g)\). By Lemma 3, there exist a constant \(\delta _{r}\) \(\left( >0\right) ,\) a sequence \(% \left\{ r_{m}\right\} _{m\in \mathbb{N} },\) \(r_{m}\rightarrow +\infty\) and a set \(E_{2}\) of finite logarithmic measure such that the estimation   \[\left\vert \frac{f(z)}{f^{\left( s\right) }(z)}\right\vert \leq r_{m}^{2s} \label{3.3}, \tag{10}\] holds for all \(z\) satisfying \(\left\vert z\right\vert =r_{m}\) \(% \notin E_{2},\) \(r_{m}\rightarrow \infty\) and \(\arg z=\) \(\theta \in\) \(% \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right]\).

Set \(z=re^{i\theta },\) \(a_{s,n}=\left\vert a_{s,n}\right\vert e^{i\theta _{s}},\) \(a_{d,n}=\left\vert a_{d,n}\right\vert e^{i\theta _{d}}\), \(b_{s,n}=\left\vert b_{s,n}\right\vert e^{i\varphi }\), \(% \theta _{s}\), \(\theta _{d}\), \(\varphi \in \lbrack 0,2\pi )\), \(\theta _{s}\neq \theta _{d}.\) Then \[\ delta \left( P_{s},\theta \right) =\left\vert a_{s,n}\right\vert \cos \left( n\theta +\theta _{s}\right) , \\ \delta \left( Q_{s},\theta \right) =\left\vert b_{s,n}\right\vert \cos \left( n\theta +\varphi \right) , \\ \delta \left( P_{d},\theta \right) =\left\vert a_{d,n}\right\vert \cos \left( n\theta +\theta _{d}\right) . \tag{11}\] Since \(a_{j,n}=c_{j}a_{s,n},\) \(b_{j,n}=c_{j}^{\prime }b_{s,n},(0<c_{j}<1),(0<c_{j}^{\prime }<1),(j\neq s)\) and \(% c_{j},c_{j}^{\prime }\) \(\left( j=0,1,…,k-1\right)\) are distinct numbers, then \[\delta \left( P_{j},\theta \right) =c_{j}\delta \left( P_{s},\theta \right) \text{ or }\delta \left( P_{j},\theta \right) =c_{j}\delta \left( P_{d},\theta \right) ,\text{ }\delta \left( Q_{j},\theta \right) =c_{j}^{\prime }\delta \left( Q_{s},\theta \right). \tag{12}\] Set \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\}\) and \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}.\) For any given \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},% \mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\right)\), we have \(\delta \left( P_{s},\theta \right) \neq 0,\) \(\delta \left( P_{d},\theta \right) \neq 0\) and \[\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) \text{ or }\delta \left( P_{s},\theta \right) <\delta \left( P_{d},\theta \right) .\]

1. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right)\). Here we also divide our proof in three subcases:  \(\left( \varphi =\theta _{s}\right)\) or  \(\left( \varphi =\theta _{d}\right)\) or  (\(\varphi \neq \theta _{s}\) and \(\varphi \neq \theta _{d}\)).

   Subcase 1.1. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) >0,\delta \left( Q_{s},\theta \right) >0.\) If \(% \delta \left( P_{s},\theta \right) >0\), \(\delta \left( Q_{s},\theta \right) >0,\) then we suppose \(\delta \left( P_{s},\theta \right) >\delta \left( Q_{s},\theta \right)\) without loss of generality. Set \(\delta _{3}=\max \left\{ \delta \left( P_{j},\theta \right) ,\delta \left( Q_{j},\theta \right) ;j\neq s\right\}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \},\) then \(0<\delta _{3}<\delta \left( P_{s},\theta \right) .\) Thus, by Lemma 4, for any given \[0<2\varepsilon <\min \left\{ \frac{\delta \left( P_{s},\theta \right) -\delta _{3}}{\delta \left( P_{s},\theta \right) +\delta _{3}},\frac{\delta \left( P_{s},\theta \right) -\delta \left( Q_{s},\theta \right) }{\delta \left( P_{s},\theta \right) +\delta \left( Q_{s},\theta \right) }\right\} ,\] where \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s\right\} ,\text{ }% c_{s}=1,\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right]\) \[\begin{align} \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }+A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert &\geq \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }\right\vert -\left\vert A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert\notag\\ &\geq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} -\exp \left\{ \left( 1+\varepsilon \right) \delta \left( Q_{s},\theta \right) r^{n}\right\}\notag\\ &\geq \frac{1}{2}\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} , \end{align}\tag{13}\] \[\begin{align} \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert &\leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ &\leq \exp \left\{ \left( 1+\varepsilon \right) \delta \left( P_{j},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1+\varepsilon \right) \delta \left( Q_{j},\theta \right) r^{n}\right\}\notag\\ &\leq 2\exp \left\{ \left( 1+\varepsilon \right) \delta _{3}r^{n}\right\} ,% \text{ }j=0,1,2,…,k-1,\text{ }j\neq s. \end{align}\tag{14}\]

   From (3), we have \[\left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }+A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert \leq \left\vert \frac{f}{f^{\left( s\right) }}\right\vert \left( \left\vert \frac{f^{\left( k\right) }}{f}\right\vert +\underset{j=0,j\neq s}{\overset{% k-1}{\sum }}\left\{ \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert \left\vert \frac{f^{\left( j\right) }}{f}\right\vert \right\} \right) . \tag{15}\] By substituting (9),(10),(13),(14), into (15), for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup \left( E_{1}\cup E_{2}\cup E_{3}\right) ,\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\) \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},% \mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\) we have \[\frac{1}{2}\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 2kBr_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) \right] ^{k+1}\exp \left\{ \left( 1+\varepsilon \right) \delta _{3}r_{m}^{n}\right\}\] which gives \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}\exp \left\{ \left( 1+\varepsilon \right) \delta _{3}r_{m}^{n}\right\} .\] Since \(0<2\varepsilon <\frac{\delta \left( P_{s},\theta \right) -\delta _{3}% }{\delta \left( P_{s},\theta \right) +\delta _{3}},\) then we can get \[\ exp \left\{ \frac{\delta \left( P_{s},\theta \right) -\delta _{3}}{2}% r_{m}^{n}\right\} \leq 4kBr_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}. \tag{16}\] By Lemma 5 and (16), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] By Lemma 6 and Eq. (3), we have \(\rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.2. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) >0,\delta \left( Q_{s},\theta \right) <0.\) We have \(\delta \left( Q_{s},\theta \right) <\delta \left( P_{s},\theta \right)\) and \(\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <0<\delta \left( P_{s},\theta \right) ,\) Put \[d=\max \left\{ c_{j}:j=0,1,…,k-1,j\neq s\right\} ,\text{ }d_{s}=1.\] By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\frac{1}{2}% \left( \frac{1-d}{1+d}\right) ),\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have \[\begin{align} \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }+A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert &\geq \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }\right\vert -\left\vert A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert\notag\\ &\geq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} -\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r^{n}\right\}\notag\\ &\geq \frac{1}{2}\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} , \end{align}\tag{17}\] \[\begin{align} \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert &\leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ &\leq \exp \left\{ \left( 1+\varepsilon \right) \delta \left( P_{j},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{j},\theta \right) r^{n}\right\}\notag\\ &\leq 2\exp \left\{ \left( 1+\varepsilon \right) d\text{ }\delta \left( P_{s},\theta \right) r^{n}\right\} ,\text{ }j=0,1,2,…,k-1,j\neq s. \end{align}\tag{18}\]

   By substituting (9),(10),(17),(18), into(15), for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup \left( E_{1}\cup E_{2}\cup E_{3}\right) ,\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right]\) \(\diagdown (H_{1}\cup H_{2}\cup H_{3}),\) we have \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}\exp \left\{ \left( 1+\varepsilon \right) d\text{ }\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} .\] Since \(0<\varepsilon <\frac{1}{2}\left( \frac{1-d}{1+d}\right) ,\) then the last inequalities leads to \[\ exp \left\{ \frac{\left( 1-d\right) }{2}\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}. \tag{19}\] By Lemma 5 and (19), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from Eq. (3), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.3. \(\delta \left( P_{s},\theta \right) >0>\delta \left( P_{d},\theta \right) ,\delta \left( Q_{s},\theta \right) >0.\) We suppose \(\delta \left( P_{s},\theta \right) >\delta \left( Q_{s},\theta \right)\) without loss of generality. By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{% 1-\nu }{1+\nu }\right) ,\frac{1}{2}\left( \frac{\delta \left( P_{s},\theta \right) -\delta \left( Q_{s},\theta \right) }{\delta \left( P_{s},\theta \right) +\delta \left( Q_{s},\theta \right) }\right) \right\} ),\) where \[\nu =\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,\text{ }\nu _{s}=1,\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},\) \(r\rightarrow +\infty\) and \[\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have \[\begin{align} \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }+A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert& \geq \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }\right\vert -\left\vert A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert\notag\\ &\geq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} -\exp \left\{ \left( 1+\varepsilon \right) \delta \left( Q_{s},\theta \right) r^{n}\right\}\notag\\ &\geq \frac{1}{2}\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} , \end{align}\tag{20}\] \[\begin{align} \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert& \leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ &\leq \exp \left\{ \left( 1+\varepsilon \right) c_{j}\delta \left( P_{s},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1+\varepsilon \right) c_{j}^{\prime }\delta \left( Q_{s},\theta \right) r^{n}\right\}\notag\\ &\leq 2\exp \left\{ \left( 1+\varepsilon \right) \nu \delta \left( P_{s},\theta \right) r^{n}\right\} ,\text{ }j=0,1,2,…,k-1,\text{ }j\neq s. \end{align}\tag{21}\]

   By substituting (9),(10),(20) and (21),into (15), for all \(% z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\)  \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right]\) \(\diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) we have \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}\exp \left\{ \left( 1+\varepsilon \right) \nu \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} .\] Since \(0<\varepsilon <\frac{1}{2}\left( \frac{1-\nu }{1+\nu }\right) ,\) then \[\ exp \left\{ \frac{\left( 1-\nu \right) }{2}\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2s}\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}. \tag{22}\] By Lemma 5 and (22), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, then by Lemma 6 and from Eq. (3), we have \(\rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.4. \(\delta \left( P_{s},\theta \right) >0>\delta \left( P_{d},\theta \right) ,\delta \left( Q_{s},\theta \right) <0.\) We have \(\delta \left( P_{s},\theta \right) >\delta \left( P_{j},\theta \right) >0,\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <0,\) then \(\delta \left( Q_{s},\theta \right) <\) \(% \delta \left( P_{s},\theta \right) .\) Put \[\nu ^{\prime }=\max \left\{ c_{j}:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,\nu _{s}^{\prime }=1.\] By Lemma 3, for any given \(0<\varepsilon <\frac{1}{2}\left( \frac{1-\nu ^{\prime }}{1+\nu ^{\prime }}\right) ,\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have \[ \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }+A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert \geq \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }\right\vert -\left\vert A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert\notag\\ \geq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} -\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r^{n}\right\}\notag\\ \geq \frac{1}{2}\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} , \tag{23}\] \[\begin{align} \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert &\leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ &\leq \exp \left\{ \left( 1+\varepsilon \right) \delta \left( P_{j},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{j},\theta \right) r^{n}\right\}\notag\\ &\leq 2\exp \left\{ \left( 1+\varepsilon \right) \nu ^{\prime }\text{ }\delta \left( P_{s},\theta \right) r^{n}\right\} ,\text{ }j=0,1,2,…,k-1,j\neq s. \end{align}\tag{24}\]By substituting (9),(10),(23),(24) into(15), for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup \left( E_{1}\cup E_{2}\cup E_{3}\right) ,\) \(r_{m}\rightarrow +\infty\) and  \[\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] we have \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2k}\left[ T\left( 2r_{m},f\right) \right] ^{k+1}\exp \left\{ \left( 1+\varepsilon \right) \nu ^{\prime }\text{ }\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} .\] Since \(0<\varepsilon <\frac{1}{2}\left( \frac{1-\nu ^{\prime }}{1+\nu ^{\prime }}\right) ,\) then \[\exp \left\{ \frac{\left( 1-\nu ^{\prime }\right) }{2}\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2k}\text{ }\left[ T\left( 2r_{m},f\right) \right] ^{k+1}. \tag{25}\] By Lemma 5 and (25) we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from Eq. (3), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.5. \(0>\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) ,\delta \left( Q_{s},\theta \right) >0.\) We have \(\delta \left( P_{d},\theta \right) <\delta \left( P_{s},\theta \right) <0<\delta \left( Q_{s},\theta \right) .\) Put \[d^{\prime }=\max \left\{ c_{j}^{\prime }:j=0,1,…,k-1,j\neq s\right\} ,% \text{ }d_{s}^{\prime }=1.\] By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\frac{1}{2}% \left( \frac{1-d^{\prime }}{1+d^{\prime }}\right) ),\) there is a set \(% E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have \[\begin{align} \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }+A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert &\geq \left\vert A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert -\left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }\right\vert\notag\\ &\geq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r^{n}\right\} -\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\}\notag\\ &\geq \frac{1}{2}\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r^{n}\right\} , \end{align}\tag{26}\] \[\begin{align} \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert &\leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ &\leq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{j},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1+\varepsilon \right) \delta \left( Q_{j},\theta \right) r^{n}\right\}\notag\\ &\leq 2\exp \left\{ \left( 1+\varepsilon \right) d^{\prime }\delta \left( Q_{s},\theta \right) r^{n}\right\} ,\text{ }j=0,1,2,…,k-1,\text{ }j\neq s. \end{align}\tag{27}\] By using a similar proof as that of subcase 1.2\(,\) since \(0<\varepsilon <% \frac{1}{2}\left( \frac{1-d^{\prime }}{1+d^{\prime }}\right) ,\) we can obtain for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and  \(\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }% \theta _{r}+\delta _{r}\right]\) \[\exp \left\{ \frac{\left( 1-d^{\prime }\right) }{2}\delta \left( Q_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) \right] ^{k+1}. \tag{28}\] So, by Lemma 5 and (28) we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from Eq. (3), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.6. \(0>\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) ,\delta \left( Q_{s},\theta \right) <0.\) Set \[\lambda =\min \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} .\] By Lemma 3, for any given \(0<\varepsilon <1,\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have \[\ \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert \leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ \leq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{j},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{j},\theta \right) r^{n}\right\}\notag\\ \leq 2\exp \left\{ \left( 1-\varepsilon \right) \lambda \delta \left( P_{s},\theta \right) r^{n}\right\} ,\text{ }j=0,1,2,…,k-1. \tag{29}\] From (3), we have \[\ 1\leq \left\vert \frac{f}{f^{\left( k\right) }}\right\vert \underset{j=0}{% \overset{k-1}{\sum }}\left\{ \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert \left\vert \frac{f^{\left( j\right) }}{f}\right\vert \right\} . \tag{30}\] By substituting (9),(10),(29) into (30), for all \(z\) satisfying \(% |z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(% r_{m}\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\) we have \[1\leq 2kBr_{m}^{2k}\left[ T\left( 2r_{m},f\right) \right] ^{k+1}\exp \left\{ \left( 1-\varepsilon \right) \lambda \delta \left( P_{s},\theta \right) r_{m}^{n}\right\}\] which gives \[\exp \left\{ \left( \varepsilon -1\right) \lambda \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 2kBr_{m}^{2k}\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}. \tag{31}\] Since \(0<\varepsilon <1\) and \(\lambda \delta \left( P_{s},\theta \right) <0,\) then by Lemma 2.5 and (31), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from Eq. (3), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 2.1. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) >0.\) Because of \(a_{s,n}\neq b_{s,n}\), we suppose \(\left\vert a_{s,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by \(\left( 3.4\right)\) and \(% \left( 3.5\right)\) we have \(\delta \left( Q_{s},\theta \right) >\delta \left( Q_{j},\theta \right) >0\), \(\delta \left( Q_{s},\theta \right) >\delta \left( P_{s},\theta \right) >\delta \left( P_{j},\theta \right) >0\). Put \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,c_{s}=1.\] Then, \(0<c<1.\) By Lemma 3, for any given \(\varepsilon\) with \[0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-c}{1+c}\right) ,\frac{1% }{2}\left( \frac{\delta \left( Q_{s},\theta \right) -\delta \left( P_{s},\theta \right) }{\delta \left( Q_{s},\theta \right) +\delta \left( P_{s},\theta \right) }\right) \right\} ,\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \[\arg z=\ \theta \in \ \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have \[ \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }+A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert \geq \left\vert A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert -\left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }\right\vert\notag\\ \geq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r^{n}\right\} -\exp \left\{ \left( 1+\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\}\notag\\ \geq \frac{1}{2}\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r^{n}\right\} , \tag{32}\] \[ \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert \leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ \leq \exp \left\{ \left( 1+\varepsilon \right) c\delta \left( P_{s},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1+\varepsilon \right) c\delta \left( Q_{s},\theta \right) r^{n}\right\}\notag\\ \leq 2\exp \left\{ \left( 1+\varepsilon \right) c\delta \left( Q_{s},\theta \right) r^{n}\right\} ,\text{ }j=0,1,2,…,k-1,\text{ }j\neq s. \tag{33}\] By substituting (9),(10),(32),(33) into (15), since \(% 0<\varepsilon <\frac{1}{2}\left( \frac{1-c}{1+c}\right) ,\) for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\)  \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\) we obtain \[\exp \left\{ \frac{\left( 1-c\right) }{2}\delta \left( Q_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}. \tag{34}\] Thus, by Lemma 5 and (34) we get \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from Eq. (3), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 2.2. \(\delta \left( P_{s},\theta \right) >0>\delta \left( P_{d},\theta \right) .\) Because of \(a_{s,n}\neq b_{s,n}\), we suppose \(\left\vert a_{s,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by (11)and(12) we have \(\delta \left( Q_{s},\theta \right) >\delta \left( Q_{j},\theta \right) >0\), \(\delta \left( Q_{s},\theta \right) >\delta \left( P_{s},\theta \right) >\delta \left( P_{j},\theta \right) >0\). Put \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,c_{s}=1.\] Using the same reasoning as in Subcase 2.1 , we can also obtain \(% \rho \left( f\right) =+\infty\) and \(\rho _{2}\left( f\right) =n.\)

   Subcase 2.3. \(0>\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) .\) We have \(\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <\delta \left( P_{s},\theta \right) <0,\) \(\delta \left( P_{s},\theta \right) <\delta \left( P_{j},\theta \right) <0\). Put \[\lambda =\min \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} .\] By Lemma 3, for any given \(0<\varepsilon <1,\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have \[ \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert ;\leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ \leq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{j},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{j},\theta \right) r^{n}\right\}\notag\\ \leq 2\exp \left\{ \left( 1-\varepsilon \right) \lambda \delta \left( P_{s},\theta \right) r^{n}\right\} ,\text{ }j=0,1,2,…,k-1. \tag{35}\] By substituting (9),(10)and(35)into(30), for all \(z\) satisfying \(% |z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(% r_{m}\rightarrow +\infty\) and \(\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\) we have \[1\leq 2kBr_{m}^{2k}\left[ T\left( 2r_{m},f\right) \right] ^{k+1}\exp \left\{ \left( 1-\varepsilon \right) \lambda \delta \left( P_{s},\theta \right) r_{m}^{n}\right\}\] which gives \[\exp \left\{ \left( \varepsilon -1\right) \lambda \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 2kBr_{m}^{2k}\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}. \tag{36}\] Since \(0<\varepsilon <1\) and \(\lambda \delta \left( P_{s},\theta \right) <0,\) then by Lemma 5 and (36) we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from Eq. (3), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 3.1. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) >0.\) Because of \(a_{d,n}\neq b_{s,n}\), we suppose \(\left\vert a_{d,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by (11) and (12), we have \(\delta \left( Q_{s},\theta \right) >\delta \left( Q_{j},\theta \right) >0\), \(\delta \left( Q_{s},\theta \right) >\delta \left( P_{s},\theta \right) >\delta \left( P_{j},\theta \right) >0\). Then, \(% 0<c<1.\) By Lemma 3, for any given \(\varepsilon\) with \[0<\varepsilon <\min \frac{1}{2}\left\{ \left( \frac{1-c}{1+c}\right) ,\frac{1% }{2}\left( \frac{\delta \left( Q_{s},\theta \right) -\delta \left( P_{s},\theta \right) }{\delta \left( Q_{s},\theta \right) +\delta \left( P_{s},\theta \right) }\right) \right\} ,\] where \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,c_{s}=1,\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \[\arg z=\ \theta \in \ \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have (32)and(33) hold. By substituting (9),(10),(32),(33)into(15), we obtain (34) for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\)  \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}).\) Since \(0<\varepsilon <\frac{1}{2}\left( \frac{1-c}{1+c}\right)\), then by Lemma 5 and (34) we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from Eq. (3), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 3.2. \(\delta \left( P_{s},\theta \right) >0>\delta \left( P_{d},\theta \right) .\) Because of \(a_{d,n}\neq b_{s,n}\), we suppose \(\left\vert a_{d,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by (11)and(12), we have \(\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <0,\) \(\delta \left( Q_{s},\theta \right) <\delta \left( P_{d},\theta \right) <0<\delta \left( P_{s},\theta \right)\). Then, \(% 0<c<1.\) By Lemma 3, for any given \(\varepsilon\) \(\left( 0<\varepsilon <% \frac{1}{2}\left( \frac{1-c}{1+c}\right) \right) ,\) where \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,c_{s}=1,\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \[\arg z=\ \theta \in \ \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have \[\begin{align} \left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }+A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert &\geq \left\vert A_{s,2}\left( z\right) e^{P_{s}\left( z\right) }\right\vert -\left\vert A_{s,1}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert\notag\\ &\geq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} -\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r^{n}\right\}\notag\\ &\geq \frac{1}{2}\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r^{n}\right\} , \end{align}\tag{37}\] \[\begin{align} \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert &\leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ &\leq \exp \left\{ \left( 1+\varepsilon \right) c\delta \left( P_{s},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1-\varepsilon \right) c\delta \left( Q_{s},\theta \right) r^{n}\right\}\notag\\ &\leq 2\exp \left\{ \left( 1+\varepsilon \right) c\delta \left( P_{s},\theta \right) r^{n}\right\} ,\text{ }j=0,1,2,…,k-1,\text{ }j\neq s. \end{align}\tag{38}\]

   By substituting (9),(10),(37)and(38)into(15), by \(% 0<\varepsilon <\frac{1}{2}\left( \frac{1-c}{1+c}\right) ,\) for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\)  \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\) we obtain \[\exp \left\{ \frac{\left( 1-c\right) }{2}\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4kBr_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}. \tag{39}\] Therefore, by Lemma 5 and (39) we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from Equation 3, we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 3.3. \(0>\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) .\) Because of \(a_{d,n}\neq b_{s,n}\), we suppose \(\left\vert a_{d,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by (11)and(12), we have \(\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <0\), \(\delta \left( Q_{s},\theta \right) <\delta \left( P_{d},\theta \right) <\delta \left( P_{s},\theta \right) <0\). Put \[c^{\prime }=\min \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1\right\} .\] By Lemma 3, for any given \(0<\varepsilon <1,\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(% H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have \[\begin{align} \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert &\leq \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }\right\vert +\left\vert A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert\notag\\ &\leq \exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{j},\theta \right) r^{n}\right\} +\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{j},\theta \right) r^{n}\right\}\notag\\ &\leq 2\exp \left\{ \left( 1-\varepsilon \right) c^{\prime }\delta \left( P_{s},\theta \right) r^{n}\right\} ,\text{ }j=0,1,2,…,k-1. \end{align}\tag{40}\]

   By substituting (9),(10)and(40)into(30), for all \(z\) satisfying \(% |z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(% r_{m}\rightarrow +\infty\) and  \[\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] we obtain \[1\leq 2kBr_{m}^{2k}\left[ T\left( 2r_{m},f\right) \right] ^{k+1}\exp \left\{ \left( 1-\varepsilon \right) c^{\prime }\delta \left( P_{s},\theta \right) r_{m}^{n}\right\}\] which gives \[\exp \left\{ \left( \varepsilon -1\right) c^{\prime }\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 2kr_{m}^{2k}B\left[ T\left( 2r_{m},f\right) \right] ^{k+1}. \tag{41}\] By Lemma 5 and (41) we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log ^{+}T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log _{2}^{+}T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from Eq. (3), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

2. \(\delta \left( P_{s},\theta \right) <\delta \left( P_{d},\theta \right)\). Here we also divide our proof in three subcases:  \(% \left( \varphi =\theta _{s}\right)\) or  \(\left( \varphi =\theta _{d}\right)\) or  (\(\varphi \neq \theta _{s}\) and \(\varphi \neq \theta _{d}\) ). Using the same reasoning as in I, we can also obtain \(\rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

4. Proof of Theorem 4

First, we show that (4) can possess at most one exceptional transcendental meromorphic solution \(f_{0}\) of finite order. In fact, if \(f_{\ast }\) is another transcendental meromorphic solution of finite order of Eq. (4), then \(% f_{0}-f_{\ast }\) is of finite order. But \(f_{0}-f_{\ast }\) is a transcendental meromorphic solution of the corresponding homogeneous equation of (4). This contradicts Theorem 3. We assume that \(f\) is an infinite order meromorphic solution of (4) whose poles are of uniformly bounded multiplicities. By Lemma 7 and Lemma 8 , we have \(\overline{\lambda }\left( f\right) =\lambda \left( f\right) =\rho \left( f\right) =+\infty\) and \(\overline{\lambda }_{2}\left( f\right) =\lambda _{2}\left( f\right) =\rho _{2}\left( f\right) \leq n.\)

Now, we prove that \(\rho _{2}\left( f\right) \geq n.\) By Lemma 3, there exists a set \(E_{1}\) \(\subset (1,+\infty )\) having finite logarithmic measure and a constant \(B>0\) such that for all \(z\) satisfying \(% |z|=r\notin \lbrack 0,1]\diagdown \lbrack E_{1}\), we have (9). Set \[\rho _{1}=\max \left\{ \rho \left( F\right) ,\rho \left( A_{j,i}(z)\right) :j=0,1,…,k-1;i=1,2\right\} .\] By (4), it follows that the poles of \(f\) can only occur at the poles of \(F\) and \(A_{j,i}(z),j=0,1,…,k-1;i=1,2\). Note that the poles of \(f\) are of uniformly bounded multiplicities. Hence \[\lambda \left( \frac{1}{f}\right) \leq \max \left\{ \rho \left( A_{j,i}(z)\right) :j=0,1,…,k-1;i=1,2\right\} =\rho _{1}.\] By Hadamard factorization theorem, we know that \(f\) can be written as \(f(z)=% \frac{g(z)}{d(z)},\) where \(g(z)\) and \(d(z)\) are entire functions with \[\lambda \left( d\right) =\rho \left( d\right) =\lambda \left( \frac{1}{f}% \right) \leq \rho _{1}<\rho \left( f\right) =\rho \left( g\right) =+\infty .\] For each sufficiently large \(|z|=r\), let \(z_{r}=re^{i\theta _{r}}\) be a point satisfying \(\left\vert g(z_{r})\right\vert\) \(=M(r,g)\). By Lemma 3, there exist a constant \(\delta _{r}\) \(\left( >0\right) ,\) a sequence \(% \left\{ r_{m}\right\} _{m\in \mathbb{N} },\) \(r_{m}\rightarrow +\infty\) and a set \(E_{2}\) of finite logarithmic measure such that the estimation (10) holds for all \(z\) satisfying \(\left\vert z\right\vert =r_{m}\) \(\notin E_{2},\) \(% r_{m}\rightarrow +\infty\) and \(\arg z=\) \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right]\). Since \(|g(z)|\) is continuous in \(|z|=r\), then there exists a constant \(r(>0)\) such that for all \(z\) satisfying \(|z|=r\) sufficiently large and \(\arg z=\) \(\theta \in\) \(% \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] ,\) we have \[\frac{1}{2}|g(z_{r})|<|g(z)|<\frac{3}{2}|g(z_{r})|. \tag{42}\] On the other hand, by Lemma 8, for a given \(\varepsilon \left( 0<\varepsilon <n-\rho _{1}\right) ,\) there exists a set \(E_{5}\subset (1,+\infty )\) that has finite linear measure and finite logarithmic measure such that when \(|z|=r\notin \lbrack 0,1]\cup E_{5},\) \(r\rightarrow +\infty ,\) we have \[\left\vert F\left( z\right) \right\vert \leq \exp \left\{ r^{\rho _{1}+\varepsilon }\right\} ,\left\vert d\left( z\right) \right\vert \leq \exp \left\{ r^{\rho _{1}+\varepsilon }\right\} . \tag{43}\] Since \(|g(z)|=M(r,g)\geq 1\), from (43), we obtain \[\left\vert \frac{F\left( z\right) }{f\left( z\right) }\right\vert =\left\vert \frac{d\left( z\right) F\left( z\right) }{g\left( z\right) }% \right\vert \leq \frac{\left\vert d\left( z\right) F\left( z\right) \right\vert }{M(r,g)} \leq \exp \left\{ r^{\rho _{1}+\varepsilon }\right\} \exp \left\{ r^{\rho _{1}+\varepsilon }\right\} =\exp \left\{ 2r^{\rho _{1}+\varepsilon }\right\}, \tag{44}\] for \(|z|=r\notin \lbrack 0,1]\cup E_{5},\) \(r\rightarrow +\infty .\) Set \(\nu =\min \left\{ \delta _{r},\lambda _{r}\right\} .\) Suppose that \(A_{j,i}(z)\), \(P_{j}(z),\) \(Q_{j}(z)\) \(a_{j,n},b_{j,n},\left( j=0,1,…,k-1;i=1,2\right)\)  satisfy the hypotheses of Theorem 3. Set \(z=re^{i\theta },\) \(% a_{s,n}=\left\vert a_{s,n}\right\vert e^{i\theta _{s}},\) \(% a_{d,n}=\left\vert a_{d,n}\right\vert e^{i\theta _{d}}\), \(b_{s,n}=\left\vert b_{s,n}\right\vert e^{i\varphi }\), \(\theta _{s}\), \(\theta _{d}\), \(\varphi \in \lbrack 0,2\pi )\), \(\theta _{s}\neq \theta _{d},.\) For any given \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}% \right] \diagdown\) \(\left( H_{1}\cup H_{2}\right)\), we have \(\delta \left( P_{s},\theta \right) \neq 0,\delta \left( P_{d},\theta \right) \neq 0\) and \[\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) \text{ or }\delta \left( P_{s},\theta \right) <\delta \left( P_{d},\theta \right) .\]

1. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right)\). Here we also divide our proof in three cases:  \(\left( \varphi =\theta _{s}\right)\) or  \(\left( \varphi =\theta _{d}\right)\) or  (\(% \varphi \neq \theta _{s}\) and \(\varphi \neq \theta _{d}\)).

   Case 1. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right)\) and (\(\varphi \neq \theta _{s}\) and \(% \varphi \neq \theta _{d}\)).

   Subcase 1.1. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) >0,\delta \left( Q_{s},\theta \right) >0.\) If \(% \delta \left( P_{s},\theta \right) >0\), \(\delta \left( Q_{s},\theta \right) >0,\) then we suppose \(\delta \left( P_{s},\theta \right) >\delta \left( Q_{s},\theta \right)\) without loss of generality. Set \(\delta _{3}=\max \left\{ \delta \left( P_{j},\theta \right) ,\delta \left( Q_{j},\theta \right) ;j\neq s\right\}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \},\) then \(0<\delta _{3}<\delta \left( P_{s},\theta \right) .\) Thus by Lemma 3, for any given \(\varepsilon\) with \[0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{\delta \left( P_{s},\theta \right) -\delta _{3}}{\delta \left( P_{s},\theta \right) +\delta _{3}}\right) ,\frac{1}{2}\left( \frac{\delta \left( P_{s},\theta \right) -\delta \left( Q_{s},\theta \right) }{\delta \left( P_{s},\theta \right) +\delta \left( Q_{s},\theta \right) }\right) ,n-\rho _{1}\right\} ,\] where \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s\right\} ,c_{s}=1,\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \[\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] we have (13)and(14) hold. From (4), we can write \[\left\vert A_{s,1}\left( z\right) e^{P_{s}\left( z\right) }+A_{s,2}\left( z\right) e^{Q_{s}\left( z\right) }\right\vert \leq \left\vert \frac{f}{f^{\left( s\right) }}\right\vert \left( \left\vert \frac{F\left( z\right) }{f}\right\vert +\left\vert \frac{f^{\left( k\right) }% }{f}\right\vert +\underset{j=0,j\neq s}{\overset{k-1}{\sum }}\left\{ \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert \left\vert \frac{f^{\left( j\right) }}{f}\right\vert \right\} \right) . \tag{45}\] By substituting (9),(10),(13),(14)and(44)into(45), for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and   \[\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] we have \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4\left( k+1\right) Br_{m}^{2s}\left[ T\left( 2r_{m},f\right) \right] ^{k+1} \exp \left\{ 2r^{\rho _{1}+\varepsilon }\right\} \exp \left\{ \left( 1+\varepsilon \right) \delta _{3}r_{m}^{n}\right\} . \tag{46}\] Since \(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{\delta \left( P_{s},\theta \right) -\delta _{3}}{\delta \left( P_{s},\theta \right) +\delta _{3}}\right) ,n-\rho _{1}\right\} ,\) then by Lemma 5 and (46), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] By Lemma 6 and from (4), we have \(\rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.2. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) >0,\delta \left( Q_{s},\theta \right) <0.\) We have \(\delta \left( Q_{s},\theta \right) <\delta \left( P_{s},\theta \right)\) and \(\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <0<\delta \left( P_{s},\theta \right) .\) Put \[d=\max \left\{ c_{j}:j=0,1,…,k-1,j\neq s\right\} ,d_{s}=1\] By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-d}{1+d}\right) ,n-\rho _{1}\right\} ),\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \[\arg z=\theta \in \ \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have (17)and(18) hold. By substituting (9),(10),(17),(18)and(44)into(45) for all \(z\) satisfying \(% |z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(% r_{m}\rightarrow +\infty\) and \(\arg z=\) \(% \theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\) we get \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4\left( k+1\right) Br_{m}^{2k}\text{ }\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}\exp \left\{ 2r^{\rho _{1}+\varepsilon }\right\} \exp \left\{ \left( 1+\varepsilon \right) d\text{ }\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} . \tag{47}\] Since \(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-d}{1+d}\right) ,n-\rho _{1}\right\} ,\) by Lemma 5 and (47), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from (4), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.3. \(\delta \left( P_{s},\theta \right) >0>\delta \left( P_{d},\theta \right) ,\delta \left( Q_{s},\theta \right) >0.\) We suppose \(\delta \left( P_{s},\theta \right) >\delta \left( Q_{s},\theta \right)\) without loss of generality. By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{% 1-\nu }{1+\nu }\right) ,\frac{1}{2}\left( \frac{\delta \left( P_{s},\theta \right) -\delta \left( Q_{s},\theta \right) }{\delta \left( P_{s},\theta \right) +\delta \left( Q_{s},\theta \right) }\right) ,n-\rho _{1}\right\} ),\) where \[\nu =\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,\nu _{s}=1,\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},\) \(r\rightarrow +\infty\) and \[\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have (20)and(21) hold. By substituting (9),(10),(20),(21)and(44)into(45) for all \(z\) satisfying \(% |z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(% r_{m}\rightarrow +\infty\) and  \[\arg z=\mathit{\ }\theta \in \mathit{\ }\left[ \theta _{r}-\delta _{r},% \mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\] we get \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4\left( k+1\right) Br_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) \right] ^{k+1} \exp \left\{ 2r^{\rho _{1}+\varepsilon }\right\} \exp \left\{ \left( 1+\varepsilon \right) \nu \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} . \tag{48}\] Since \(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-\nu }{1+\nu }% \right) ,n-\rho _{1}\right\} ,\) then by Lemma 5 and (48), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, then by Lemma 6 and from (4), we have \(\rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.4. \(\delta \left( P_{s},\theta \right) >0>\delta \left( P_{d},\theta \right) ,\delta \left( Q_{s},\theta \right) <0.\) We have \(\delta \left( P_{s},\theta \right) >\delta \left( P_{j},\theta \right) >0,\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <0,\) then \(\delta \left( Q_{s},\theta \right) <\) \(% \delta \left( P_{s},\theta \right) .\) Put \[\nu ^{\prime }=\max \left\{ c_{j}:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,\nu _{s}^{\prime }=1.\] By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-\nu ^{\prime }}{1+\nu ^{\prime }}\right) ,n-\rho _{1}\right\} ),\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin % \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \[\arg z=\theta \in \ \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have \(\left( 3.16\right)\) and \(\left( 3.17\right)\) hold. By substituting (9),(10),(23),(24)and(44)into(45) for all \(z\) satisfying \(% |z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(% r_{m}\rightarrow +\infty\) and  \[\arg z=\mathit{\ }\theta \in \mathit{\ }\left[ \theta _{r}-\delta _{r},% \mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] we have \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4\left( k+1\right) Br_{m}^{2k}\text{ }\left[ T\left( 2r_{m},f\right) % \right] ^{k+1}\exp \left\{ 2r^{\rho _{1}+\varepsilon }\right\} \exp \left\{ \left( 1+\varepsilon \right) \nu ^{\prime }\text{ }\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} . \tag{49}\] Since \(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-\nu ^{\prime }}{% 1+\nu ^{\prime }}\right) ,n-\rho _{1}\right\} ,\) then by Lemma 5 and (49), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from (4), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.5. \(0>\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) ,\delta \left( Q_{s},\theta \right) >0.\) We have \(\delta \left( P_{d},\theta \right) <\delta \left( P_{s},\theta \right) <0<\delta \left( Q_{s},\theta \right) .\) Put \[d^{\prime }=\max \left\{ c_{j}^{\prime }:j=0,1,…,k-1,j\neq s\right\} ,d_{s}^{\prime }=1.\] By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-d^{\prime }}{1+d^{\prime }}\right) ,n-\rho _{1}\right\} ),\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin % \left[ 0,1\right] \cup E_{3},\) \(r\rightarrow +\infty\) and \[\arg z=\theta \in \ \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have (26) and (27) hold. Using a similar proof as that of Subcase 1.5 of Theorem 3 by (45), we can obtain for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\)  \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right]\) \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r_{m}^{n}\right\} \leq 4\left( k+1\right) Br_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) \right] ^{k+1} \exp \left\{ 2r^{\rho _{1}+\varepsilon }\right\} \exp \left\{ \left( 1+\varepsilon \right) d^{\prime }\delta \left( Q_{s},\theta \right) r_{m}^{n}\right\} . \tag{50}\] Since \(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-d^{\prime }}{% 1+d^{\prime }}\right) ,n-\rho _{1}\right\} ,\) by Lemma 5 and (50), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from (4), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 1.6. \(0>\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) ,\delta \left( Q_{s},\theta \right) <0.\) Set \[\lambda =\min \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} .\tag{51}\] By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\min \left\{ \frac{1}{2},n-\rho _{1}\right\} ),\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},\) \(r\rightarrow +\infty\) and \(\arg z=\theta \in \ \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0,\delta \left( Q_{s},\theta \right) =\delta \left( P_{s},\theta \right) \}\) are finite sets, we have (29) holds\(.\) From (4), we have \[1\leq \left\vert \frac{f}{f^{\left( k\right) }}\right\vert \left( \left\vert \frac{F\left( z\right) }{f\left( z\right) }\right\vert +\underset{j=0}{% \overset{k-1}{\sum }}\left\{ \left\vert A_{j,1}\left( z\right) e^{P_{j}\left( z\right) }+A_{j,2}\left( z\right) e^{Q_{j}\left( z\right) }\right\vert \left\vert \frac{f^{\left( j\right) }}{f}\right\vert \right\} \right) . \tag{51}\] By substituting (9),(10),(29)and(44)into(51) for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\theta \in\) \(% \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\) we get \[1\leq 2\left( k+1\right) Br_{m}^{2k}\left[ T\left( 2r_{m},f\right) \right] ^{k+1}\exp \left\{ 2r^{\rho _{1}+\varepsilon }\right\} \exp \left\{ \left( 1-\varepsilon \right) \lambda \delta \left( P_{s},\theta \right) r_{m}^{n}\right\}\] which gives \[\exp \left\{ \left( \varepsilon -1\right) \lambda \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 2\left( k+1\right) Br_{m}^{2k}\exp \left\{ 2r^{\rho _{1}+\varepsilon }\right\} \left[ T\left( 2r_{m},f\right) \right] ^{k+1}. \tag{52}\] Since \(0<\varepsilon <\min \left\{ \frac{1}{2},n-\rho _{1}\right\}\) and \(% \delta \left( P_{s},\theta \right) <0,\) by Lemma 5 and (52), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from (4), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Case 2. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right)\) and\(\ \varphi =\theta _{s}.\mathbf{\ }\)

   Subcase 2.1. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) >0.\) Because of \(a_{s,n}\neq b_{s,n}\), we suppose \(\left\vert a_{s,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by (11) and (12), we have \(\delta \left( Q_{s},\theta \right) >\delta \left( Q_{j},\theta \right) >0\), \(\delta \left( Q_{s},\theta \right) >\delta \left( P_{s},\theta \right) >\delta \left( P_{j},\theta \right) >0\). Put \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,c_{s}=1.\] Then, \(0<c<1.\) By Lemma 3, for any given \(\varepsilon\) with \[0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-c}{1+c}\right) ,\frac{1% }{2}\left( \frac{\delta \left( Q_{s},\theta \right) -\delta \left( P_{s},\theta \right) }{\delta \left( Q_{s},\theta \right) +\delta \left( P_{s},\theta \right) }\right) ,n-\rho _{1}\right\},\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logar measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \[\arg z=\ \theta \in \ \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have (32) and (33) hold. By substituting (9),(10),(32),(33)and(44)into(45), we obtain for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\) \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},% \mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3})\) \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r_{m}^{n}\right\} \leq 4\left( k+1\right) Br_{m}^{2s}\text{ }\left[ T\left( 2r_{m},f\right) \right] ^{k+1} \exp \left\{ 2r^{\rho _{1}+\varepsilon }\right\} \exp \left\{ \left( 1+\varepsilon \right) c\delta \left( Q_{s},\theta \right) r_{m}^{n}\right\} .\tag{53}\] Since \(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-c}{1+c}\right) ,n-\rho _{1}\right\}\), then by Lemma 5 and (53) we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and (4), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 2.2. \(\delta \left( P_{s},\theta \right) >0>\delta \left( P_{d},\theta \right) .\) Because of \(a_{s,n}\neq b_{s,n}\), we suppose \(\left\vert a_{s,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by (11) and (12) we have \(\delta \left( Q_{s},\theta \right) >\delta \left( Q_{j},\theta \right) >0\), \(\delta \left( Q_{s},\theta \right) >\delta \left( P_{s},\theta \right) >\delta \left( P_{j},\theta \right) >0\). Put \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,c_{s}=1.\] Using the same reasoning as in Subcase 2.1 , we can also obtain \(% \rho \left( f\right) =+\infty\) and \(\rho _{2}\left( f\right) =n.\)

   Subcase 2.3. \(0>\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) .\) We have \(\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <\delta \left( P_{s},\theta \right) <0,\) \(\delta \left( P_{s},\theta \right) <\delta \left( P_{j},\theta \right) <0\). Put \[\lambda =\min \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} .\] By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\min \left\{ \frac{1}{2},n-\rho _{1}\right\} ),\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have (35) holds\(.\) By (9),(10),(35),(44)and(51), we have for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup \left( E_{1}\cup E_{2}\cup E_{3}\right) ,\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown (H_{1}\cup H_{2}\cup H_{3}),\) \[1\leq 2\left( k+1\right) Br_{m}^{2k}\left[ T\left( 2r_{m},f\right) \right] ^{k+1}\exp \left( 2r^{\rho _{1}+\varepsilon }\right) \exp \left\{ \left( 1-\varepsilon \right) \lambda \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} ,\] which gives \[\exp \left\{ \left( \varepsilon -1\right) \lambda \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 2\left( k+1\right) Bkr_{m}^{2k}\exp \left( 2r^{\rho _{1}+\varepsilon }\right) \left[ T\left( 2r_{m},f\right) \right] ^{k+1}. \tag{54}\] Since \(0<\varepsilon <n-\rho _{1}\) and \(\delta \left( P_{s},\theta \right) <0,\) then by Lemma 5 and (54), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from (4), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Case 3. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right)\) and\(\ \varphi =\theta _{d}.\ \)

   Subcase 3.1. \(\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) >0.\) Because of \(a_{d,n}\neq b_{s,n}\), we suppose \(\left\vert a_{d,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by (11)and(12), we have \(\delta \left( Q_{s},\theta \right) >\delta \left( Q_{j},\theta \right) >0\), \(\delta \left( Q_{s},\theta \right) >\delta \left( P_{s},\theta \right) >\delta \left( P_{j},\theta \right) >0\). Then, \(0<c<1.\) By Lemma 3, for any given \(\varepsilon\) with \[0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-c}{1+c}\right) ,\frac{1% }{2}\left( \frac{\delta \left( Q_{s},\theta \right) -\delta \left( P_{s},\theta \right) }{\delta \left( Q_{s},\theta \right) +\delta \left( P_{s},\theta \right) }\right) ,n-\rho _{1}\right\} ,\] where \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,c_{s}=1,\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},\) \(r\rightarrow +\infty\) and \[\arg z=\theta \in \left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\] where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have (32)and(33) hold. By substituting (9),(10),(32),(33)and(44)into(45), we obtain \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( Q_{s},\theta \right) r_{m}^{n}\right\} \leq 4\left( k+1\right) Br_{m}^{2s}\left[ T\left( 2r_{m},f\right) \right] ^{k+1} \exp \left( 2r^{\rho _{1}+\varepsilon }\right) \exp \left\{ \left( 1+\varepsilon \right) c\delta \left( Q_{s},\theta \right) r_{m}^{n}\right\} \tag{55}\] for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\)  \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) .\) Since \(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{% 1-c}{1+c}\right) ,n-\rho _{1}\right\}\), then by Lemma 5 and (55), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from (4), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 3.2. \(\delta \left( P_{s},\theta \right) >0>\delta \left( P_{d},\theta \right) .\) Because of \(a_{d,n}\neq b_{s,n}\), we suppose \(\left\vert a_{d,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by (11)and(12), we have \(\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <0\), \(\delta \left( Q_{s},\theta \right) <\delta \left( P_{d},\theta \right) <0<\delta \left( P_{s},\theta \right)\). Then, \(0<c<1.\) By Lemma 3, for any given \(\varepsilon\) \(\left( 0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-c}{1+c}\right) ,n-\rho _{1}\right\} \right) ,\) where \[c=\max \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1,j\neq s,j\neq d\right\} ,c_{s}=1,\] there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logar measure such that for all \(z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\) \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right]\) \(% \diagdown H_{1}\cup H_{2}\cup H_{3},\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(% H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have (37)and(38) hold. Substituting (9),(10),(37),(38)and(44)into(45), we obtain for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1\right] \cup (E_{1}\cup E_{2}\cup E_{3}),\) \(r_{m}\rightarrow +\infty\) and  \(\arg z=\) \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right)\) \[\exp \left\{ \left( 1-\varepsilon \right) \delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 4\left( k+1\right) Br_{m}^{2s}\left[ T\left( 2r_{m},f\right) \right] ^{k+1} \exp \left\{ \left( 1+\varepsilon \right) c\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} . \tag{56}\] Since \(0<\varepsilon <\min \left\{ \frac{1}{2}\left( \frac{1-c}{1+c}\right) ,n-\rho _{1}\right\}\), then by Lemma 5 and (56), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from (4), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

   Subcase 3.3. \(0>\delta \left( P_{s},\theta \right) >\delta \left( P_{d},\theta \right) .\) Because of \(a_{d,n}\neq b_{s,n}\), we suppose \(\left\vert a_{d,n}\right\vert <\left\vert b_{s,n}\right\vert\) without loss of generality. In this case, by (11) and (12), we have \(\delta \left( Q_{s},\theta \right) <\delta \left( Q_{j},\theta \right) <0\), \(\delta \left( Q_{s},\theta \right) <\delta \left( P_{d},\theta \right) <\delta \left( P_{s},\theta \right) <0\). Put \[c^{\prime }=\min \left\{ c_{j},c_{j}^{\prime }:j=0,1,…,k-1\right\} .\] By Lemma 3, for any given \(\varepsilon\) (\(0<\varepsilon <\min \left\{ \frac{1}{2},n-\rho _{1}\right\} ),\) there is a set \(E_{3}\subset \lbrack 1,+\infty )\) having finite logarithmic measure such that for all \(% z\) satisfying \(|z|=r\notin \left[ 0,1\right] \cup E_{3},r\rightarrow +\infty\) and \(\arg z=\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{% \ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right) ,\) where \(H_{1}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =0,\delta \left( P_{d},\theta \right) =0\},\) \(% H_{2}=\{\theta \in \lbrack 0,2\pi ):\delta \left( P_{s},\theta \right) =\delta \left( P_{d},\theta \right) \}\) and \(H_{3}=\{\theta \in \lbrack 0,2\pi ):\delta \left( Q_{s},\theta \right) =0\}\) are finite sets, we have (40) holds\(.\) By substituting (9),(10),(40)and(44)into(51), we obtain for all \(z\) satisfying \(|z|=r_{m}\notin \left[ 0,1% \right] \cup \left( E_{1}\cup E_{2}\cup E_{3}\right) ,\) \(r_{m}\rightarrow +\infty\) and \(\arg z=\) \(\theta \in\) \(\left[ \theta _{r}-\delta _{r},\mathit{\ }\theta _{r}+\delta _{r}\right] \diagdown \left( H_{1}\cup H_{2}\cup H_{3}\right)\) \[1\leq 2\left( k+1\right) Br_{m}^{2k}\left[ T\left( 2r_{m},f\right) \right] ^{k+1}\exp \left( 2r^{\rho _{1}+\varepsilon }\right) \exp \left\{ \left( 1-\varepsilon \right) c^{\prime }\delta \left( P_{s},\theta \right) r_{m}^{n}\right\}\] which gives \[\exp \left\{ \left( \varepsilon -1\right) c^{\prime }\delta \left( P_{s},\theta \right) r_{m}^{n}\right\} \leq 2\left( k+1\right) Br_{m}^{2k} \left[ T\left( 2r_{m},f\right) \right] ^{k+1}\exp \left( 2r^{\rho _{1}+\varepsilon }\right) . \tag{57}\] By Lemma 5 and (57), we obtain \[\rho \left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log T\left( r_{m},f\right) }{\log r_{m}}=+\infty\] and \[\rho _{2}\left( f\right) =\limsup_{r_{m}\rightarrow +\infty }\frac{\log \log T\left( r_{m},f\right) }{\log r_{m}}\geq n.\] In addition, by Lemma 6 and from (4), we have \(% \rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)

2. \(\delta \left( P_{s},\theta \right) <\delta \left( P_{d},\theta \right)\). Here we also divide our proof in three subcases:  \(% \left( \varphi =\theta _{s}\right)\) or  \(\left( \varphi =\theta _{d}\right)\) or  (\(\varphi \neq \theta _{s}\) and \(\varphi \neq \theta _{d}\) ). Using the same reasoning as in I, we can obtain \(\rho _{2}\left( f\right) \leq n,\) so \(\rho _{2}\left( f\right) =n.\)