On a class of \(p\)-valent functions with negative coefficients defined by opoola differential operator

Author(s): Bitrus Sambo1, Timothy Oloyede Opoola2
1Department of Mathematics, Gombe State University, P.M.B. 127, Gombe, Nigeria.
2Department of Mathematics, University of Ilorin, P.M.B. 1515, Ilorin, Nigeria.
Copyright © Bitrus Sambo, Timothy Oloyede Opoola. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using opoola differential operator, we defined a subclass \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) of the class of multivalent or p-valent functions. Several properties of the class were studied, such as coefficient inequalities, hadamard product, radii of close-to-convex, star-likeness, convexity, extreme points, the integral mean inequalities for the fractional derivatives, and further growth and distortion theorem are given using fractional calculus techniques.

Keywords: Multivalent functions; Opoola differential operator; Coefficient inequalities; Closure property.

1. Introduction

Let \(A\) denote the class of all functions, \(f(z)\) normalized by
\begin{align}\tag{1}\label{eq1.1} f(z)=z+\sum_{k=2}^{\infty}a_{k}z^{k}\,, \end{align}
which are analytic in the unit disc \(U=\lbrace z:|z|< 1\rbrace\).

Definition 3. [1] For \(f(z)\in A\) , Opoola introduced the following operator:

\begin{align}\tag{2}\label{eq1.2} D^{0}(\mu,\beta,t)f(z)&=f(z),\notag\\ D^{1}(\mu,\beta,t)f(z)&=zD_{t}f(z)=tzf'(z)-{z(\beta-\mu)t}+[1+(\beta-\mu-1)t]f(z),\notag\\ D^{n}(\mu,\beta,t)f(z)&=zD_{t} (D^{n-1}(\mu,\beta,t)f(z)),& n\in N. \end{align}
If \(f(z)\) is given by (\ref{eq1.1}), then from (2), we see that
\begin{align}\tag{3}\label{eq1.3} D^{n}(\mu,\beta,t)f(z)=z+\sum_{k=2}^{\infty}\left[1+\left(k+\beta-\mu-1\right) t\right] ^{n}a_kz^{k}\,, \end{align}
\( (0\leq\mu \leq \beta\), \(t\ge 0\) and \(n\in N_{0}=N \cup{0})\).

Remark 1.

  1. When \(\beta=\mu\), \(t=1\),\( D^{n}(\mu,\beta,t)f(z)= D^{n}f(z)\) by Salagean [2],
  2. When \(\beta=\mu\),\( D^{n}(\mu,\beta,t)f(z)= D_{\lambda}^{n}f(z)\) by Al-Oboudi [3].

Definition 1. Let \(A_p\) denote the class of functions of the form:

\begin{align}\tag{4}\label{eq1.4} f(z)=z^{p} + \sum_{k=p+1}^{\infty}a_kz^{k},\;\;\;\;\;(p=1,2,…)\,, \end{align}
which are analytic and multivalent in the open unit disc \(U=({z\in C:\left|z\right|< 1 }).\) We define the following differential operator for the functions \(f(z)\in A_p\)
\begin{align}\tag{5} D^{0}(\mu,\beta,t,p)f(z)& =f(z),\notag\\ D^{1}(\mu,\beta,t,p)f(z)& =zD_{t,p}f(z)=\frac{t}{p}zf'(z)-z^{p}(\beta-\mu)t+[1+(\beta-\mu-1)t]f(z),\notag\\ D^{n}(\mu,\beta,t,p)f(z)& =zD_{t,p} (D^{n-1}(\mu,\beta,t,p)f(z)), & n\in N.\label{eq1.5} \end{align}
If \(f(z)\) is given by (\ref{eq1.4}), then from (5), we see that
\begin{align}\tag{6}\label{eq1.6} D^{n}(\mu,\beta,t,p)f(z)=z^{p}+\sum_{k=p+1}^{\infty}\left[ 1+\left( \frac{k}{p}+\beta-\mu-1\right)t\right]^{n}a_kz^{k}\,, \end{align}
\( (0\leq\mu \leq \beta\), \(t\ge 0\) and \(n\in N_{0}=N \cup{0})\). Let \(T_{p}\) denote the subclass of \(A_{p}\) consisting of functions of the form
\begin{align}\tag{7}\label{eq1.7} f(z)=z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k} ,&& (a_k\ge 0,p=1,2,…)\,. \end{align}
If \(f(z)\) is given by Eq. (\ref{eq1.7}), then from Eq. (5), we get
\begin{align}\tag{8}\label{eq1.8} D^{n}(\mu,\beta,t,p)f(z)=z^{p}-\sum_{k=p+1}^{\infty}\left[ 1+\left( \frac{k}{p}+\beta-\mu-1\right)t\right]^{n}a_kz^{k}, \end{align}
\((n\in N_{0},a_k\ge 0,p=1,2,..,0\leq\mu \leq \beta, t\ge 0,n\in N_{0}=N \cup{0})\,.\)

Remark 2. When \(\beta=\mu\) in (\ref{eq1.8}),\(D^{n}(\mu,\beta,t,p)f(z)=D^{n}_{\delta,p}f(z)\) defined by Bulut in [4]. Now, from (\ref{eq1.8}), it follows that \(D^{n}(\mu,\beta,t,p)f(z)\) can be written in terms of Convolution as $$D^{n}(\mu,\beta,t,p)f(z)=(f\ast g)(z)\,,$$ where \(f(z)\) is as in (\ref{eq1.7}), while $$g(z)=z^{p}-\sum_{k=p+1}^{\infty}\left[ 1+\left( \frac{k}{p}+\beta-\mu-1\right)t\right]^{n}z^{k}\,.$$

Definition 2. A function \(f(z)\in T_{p}\) is in the class \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) if

\begin{align}\tag{9}\label{eq1.9} \left|\frac{(D^{n}(\mu,\beta,t,p)f(z))’-pz^{p-1}}{\lambda (D^{n}(\mu,\beta,t,p)f(z))’+(\alpha-\gamma)}\right|< \delta, && (z\in U,n\in N_{0})\,, \end{align}
for some \(0\le\lambda< 1\), \(0\le\gamma< 1\),\(0< \alpha\le 1\),\(0< \delta< 1\), \(D^{n}(\mu,\beta,t,p)f(z)\) as defined in (\ref{eq1.8}).

Remark 3. When \(\mu=\beta\) in (\ref{eq1.9}),the class \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) reduces to the class \(R^{n}_{p}(\alpha,\beta,\gamma,\mu)\) studied by Bulut in [4].

Definition 4. [5, 6] The fractional integral of order \(l\) is defined, for function \(f(z)\) by

\begin{align}\tag{10}\label{eq1.10} D_{z}^{-l}f(z)=\frac{1}{\Gamma (l)}\int_{0}^{z}\frac{f(t)}{(z-t)^{1-l}}d(t),&&(l>0)\,, \end{align}
where \(f(z)\) is an analytic function in a simply connected region of \(z\)-plane containing the origin,and the multiplicity of \((z-t)^{l-1}\) is removed by requiring \(log(z-t)\) to be real when \((z-t)>0\).

Definition 5. [5, 6] The fractional derivative of order \(l\) is defined, for function \(f(z)\) by

\begin{align}\tag{11}\label{eq1.11} D_{z}^{l}f(z)=\frac{1}{\Gamma (1-l)}\frac{d}{d(z)}\int_{0}^{z}\frac{f(t)}{(z-t)^{l}}d(t),& & (0\le l< 1)\,, \end{align}
where \(f\) is an analytic function in a simply connected region of \(z\)-plane containing the origin,and the multiplicity of \((z-t)^{-l}\) is removed by requiring \(log(z-t)\) to be real when \((z-t)>0\).

Definition 6. [5, 6] Under the hypothesis of Definition 4, the fractional derivative of order \(p+l\) is defined for functions \(f(z)\), by

\begin{align}\tag{12}\label{eq1.12} D_{z}^{p+l}f(z)=\frac{d^{p}}{d(z)^{p}}D_{z}^{l}f(z), & &(0\le l< 1,p\in N_{0})\,. \end{align}
It readily follows from (\ref{eq1.9}) and (\ref{eq1.10}) that
\begin{align}\tag{13}\label{eq1.13} D_{z}^{-l}z^{k}=\frac{\Gamma (k+1)}{\Gamma (k+l+1)}z^{k+l},& & (l>0,k\in N)\,, \end{align}
and
\begin{align}\tag{14}\label{eq1.14} D_{z}^{l}z^{k}=\frac{\Gamma (k+1)}{\Gamma (k-l+1)}z^{k-l}, & & (0\le l< 1,k\in N)\,. \end{align}

Lemma 1.[7] If \(f(z)\) and \(g(z)\) are analytic in \(U\) with \(f(z) \prec g(z)\), then for \(\sigma >0\) and \(z=re^{i\theta}\), \({(0 < r< 1)}\), then \[\int_{0}^{2\pi}\left|f(z)\right|^{\sigma}d\theta \le \int_{0}^{2\pi}\left|g(z)\right|^{\sigma}d\theta\,.\]

In this work, several properties of the class \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) are studied, such as coefficient inequalities, hadamard product, radii of close-to-convex, star-likeness, convexity, extreme points, the integral mean inequalities for the fractional derivatives, and further growth and distortion theorem are given using fractional calculus techniques. For more research on classes of multivalent or p-valent functions, see [7, 8, 9, 10, 11, 12, 13,14, 15, 16, 17, 18]

2. Main results

Theorem 1. A function \(f(z)\in T_{p}\) is in the class \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) if and only if

\begin{align}\tag{15}\label{eq1.15} \sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)a_k \le \delta(\lambda p+\alpha-\gamma)\,, \end{align}
for some \(0\le\lambda< 1\), \(0\le\gamma< 1\),\(0< \alpha\le 1\),\(0< \delta< 1\). The result is sharp for the function \(f(z)\) given by \begin{align*}f(z)=z^{p}-\frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}z^{k},& & (k\ge p+1).\end{align*}

Proof Suppose that \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta) \), then we have from (\ref{eq1.9}) that $$\left|\frac{(D^{n}(\mu,\beta,t,p)f(z))’-pz^{p-1}}{\lambda (D^{n}(\mu,\beta,t,p)f(z))’+(\alpha-\gamma)}\right|< \delta\,.$$ By substitution, we have $$\left|\frac{pz^{p-1}-\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_kz^{k-1}-pz^{p-1}}{\lambda (pz^{p-1}-\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_kz^{k-1})+(\alpha-\gamma)}\right|< \delta,$$ $$\left|\frac{\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_kz^{k-1}}{\lambda (pz^{p-1}-\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_kz^{k-1})+(\alpha-\gamma)}\right|< \delta\,.$$ Since \(\Re z\le\left|z\right|,\) then $$\Re \left\lbrace\frac{\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_kz^{k-1}}{\lambda (pz^{p-1}-\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_kz^{k-1})+(\alpha-\gamma)}\right\rbrace < \delta\,. $$ If we choose \(z\) real and let \(z\rightarrow 1^{-}\), then we get \begin{align*}& \sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_k\le \delta\left[ \lambda (p-\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_k)+(\alpha-\gamma)\right],\\ & \Rightarrow \sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_k \le \delta \lambda p -\delta \lambda\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_k+\delta(\alpha-\gamma),\\ & \Rightarrow \sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_k+\delta \lambda\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_k\le\delta(\lambda p+\alpha-\gamma),\\ & \Rightarrow \sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta \lambda)a_k \le\delta(\lambda p+\alpha-\gamma).\end{align*} Conversely, suppose that the inequality (\ref{eq1.15}) holds true and that \(z\in \partial U:\left\lbrace z\in C:\left| z\right|=1\right\rbrace\) and suppose that \begin{align*}& \left|(D^{n}(\mu,\beta,t,p)f(z))'-pz^{p-1}\right| – \delta\left| \lambda (D^{n}(\mu,\beta,t,p)f(z))'+(\alpha-\gamma)\right|\\ & \le\left|(D^{n}(\mu,\beta,t,p)f(z))'-pz^{p-1}- \delta(\lambda(D^{n}(\mu,\beta,t,p)f(z))'+(\alpha-\gamma))\right|\\ & =\left|-\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_{k}z^{k-1} -\delta\lambda pz^{p-1} \right.\\&+\left. \delta\lambda\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_{k}z^{k-1}-\delta(\alpha-\gamma)\right|\\ & =\left|\sum_{k=p+1}^{\infty}(\delta\lambda-1)k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_{k}z^{k-1}-\delta(\lambda pz^{p-1}+\alpha-\gamma)\right|\\ & \le\sum_{k=p+1}^{\infty}(\delta\lambda-1)k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}a_{k}-\delta(\lambda p+\alpha-\gamma)\\ & \le\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(\delta\lambda+1)a_{k}-\delta(\lambda p+\alpha-\gamma)\le 0.\end{align*} Since by maximum modulus theorem ,that the maximum modulus of an analytic function cannot be attained inside the domain but on the boundary, implies $$\left|(D^{n}(\mu,\beta,t,p)f(z))'-pz^{p-1}\right|- \delta\left| \lambda (D^{n}(\mu,\beta,t,p)f(z))'+(\alpha-\gamma)\right|< 0\,,$$ i.e.,$$ \left|(D^{n}(\mu,\beta,t,p)f(z))'-pz^{p-1}\right|< \delta\left| \lambda (D^{n}(\mu,\beta,t,p)f(z))'+(\alpha-\gamma)\right|\,.$$ So, $$\frac{\left|(D^{n}(\mu,\beta,t,p)f(z))'-pz^{p-1}\right|}{\left|\lambda (D^{n}(\mu,\beta,t,p)f(z))'+(\alpha-\gamma)\right|}< \delta\,,$$ implies $$ \left|\frac{(D^{n}(\mu,\beta,t,p)f(z))'-pz^{p-1}}{\lambda (D^{n}(\mu,\beta,t,p)f(z))'+(\alpha-\gamma)}\right|< \delta.$$ Hence, we have that \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta).\)

Corollary 1. If \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then $$a_{p+1}\le \frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{(p+1)[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)}\,.$$

Theorem 2. The class \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) is a class of convex functions.

Proof Let the functions

\begin{align}\tag{16}\label{eq1.16} f(z)=z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k} ,&& (a_k\ge 0,p=1,2,…), \end{align}
\begin{align}\tag{17}\label{eq1.17} g(z)=z^{p}-\sum_{k=p+1}^{\infty}b_{k}z^{k} ,&& (b_k\ge 0,p=1,2,…), \end{align}
be in the class \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then for \(0\le j\le1\) $$h(z)=(1-j)f(z)+jg(z)=z^{p}-\sum_{k=p+1}^{\infty}c_{k}z^{k}\,,$$ where \(c_{k}=(1-j) a_{k}+j b_{k} \ge0\), then making use of (\ref{eq1.15}), we see that \begin{align*}\sum_{k=p+1}^{\infty}k&[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)c_k\\ &=\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)a_k+\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)b_k\\ &< (1-j)\delta(\lambda p+\alpha-\gamma)+j\delta(\lambda p+\alpha-\gamma)\\ &=\delta(\lambda p+\alpha-\gamma)-j\delta(\lambda p+\alpha-\gamma)+j\delta(\lambda p+\alpha-\gamma)\\ &=\delta(\lambda p+\alpha-\gamma),\end{align*} implies \(h(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), which completes the proof.

Theorem 3. If each of the functions \(f(z)\) and \(g(z)\) is in the class \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then \((f\ast g)(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\Omega)\), where \(\Omega \ge \frac{\delta^{2}(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)^{2}-\delta^{2}\lambda(\lambda p+\alpha-\gamma)}\).

Proof From (\ref{eq1.15}), we have

\begin{align}\tag{18}\label{eq1.18} \sum_{k=p+1}^{\infty}\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}a_{k}\le 1\,, \end{align}
and
\begin{align}\tag{19}\label{eq1.19} \sum_{k=p+1}^{\infty}\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}b_{k}\le 1\,. \end{align}
We need to find the smallest \(\Omega\) such that
\begin{align}\tag{20}\label{eq1.20} \sum_{k=p+1}^{\infty}\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\Omega\lambda)}{\Omega(\lambda p+\alpha-\gamma)}a_{k}b_{k}\le 1\,. \end{align}
From (\ref{eq1.18}) and (\ref{eq1.19}), we find by means of Cauchy-Schwarz inequalities that
\begin{align}\tag{21}\label{eq1.21} \sum_{k=p+1}^{\infty}\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\sqrt {a_{k}b_{k}}\le 1\,. \end{align}
Thus, it is enough to show that $$\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\Omega\lambda)}{\Omega(\lambda p+\alpha-\gamma)}a_{k}b_{k}\le \frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\sqrt {a_{k}b_{k}}\,.$$ That is
\begin{align}\tag{22}\label{eq1.22} \sqrt{a_{k}b_{k}} \le \frac{\Omega(1+\delta \lambda)}{\delta(1+\Omega \lambda)}\,. \end{align}
On the other hand, from (\ref{eq1.21}), we have
\begin{align}\tag{23}\label{eq1.23} \sqrt{a_{k}b_{k}} \le \frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}\,. \end{align}
Therefore, in view of (\ref{eq1.22}) and (\ref{eq1.23}), it is enough to show that $$\frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}\le \frac{\Omega(1+\delta \lambda)}{\delta(1+\Omega \lambda)},$$ i.e., $$ \delta(\lambda p+\alpha-\gamma)\delta(1+\Omega \lambda)\le kM(1+\delta\lambda)\Omega(1+\delta \lambda)\,,$$ where \(M=[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}\). So, $$\delta^{2}[(\lambda p+\alpha-\gamma)+\Omega \lambda(\lambda p+\alpha-\gamma)]\le k\Omega M(1+\delta \lambda)^{2}\,,$$ implies $$ \delta^{2}(\lambda p+\alpha-\gamma)+\delta^{2}\Omega \lambda(\lambda p+\alpha-\gamma) \le k\Omega M(1+\delta \lambda)^{2}\,,$$ implies \[ \delta^{2}(\lambda p+\alpha-\gamma) \le k\Omega M(1+\delta \lambda)^{2}-\delta^{2}\Omega \lambda(\lambda p+\alpha-\gamma)\,.\] Also \[\Omega\left[ k M(1+\delta \lambda)^{2}-\delta^{2} \lambda(\lambda p+\alpha-\gamma)\right] \ge \delta^{2}(\lambda p+\alpha-\gamma)\,,\] implies \[ \Omega \ge\frac{\delta^{2}(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)^{2}-\delta^{2}\lambda(\lambda p+\alpha-\gamma)}\,.\]

Theorem 4. If \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then \(f(z)\) is p-valently close-to-convex of order \(\rho\) in \(\left|z\right|< r_{1}(\lambda,\alpha,\gamma,\delta,\rho)\), where \begin{align*}r_{1}(\lambda,\alpha,\gamma,\delta,\rho)=\inf_{k}\left\lbrace\frac{[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)(p-\rho)}{\delta(\lambda p+\alpha-\gamma)} \right\rbrace^{\frac{1}{k-p}},& & (k\ge p+1)\,.\end{align*}

Proof Let \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then \[\left|\frac{f'(z)}{z^{p-1}}-p\right|< p-\rho\,,\] implies

\begin{align}\tag{24}\label{eq1.24} \left|\frac{pz^{p-1}-\sum_{k=p+1}^{\infty}ka_{k}z^{k-1}-pz^{p-1}}{z^{p-1}}\right|=\left|\sum_{k=p+1}^{\infty}ka_{k}z^{k-p}\right|\le\sum_{k=p+1}^{\infty}ka_{k}\left|z\right|^{k-p}< p-\rho\,. \end{align}
Since
\begin{align}\tag{25}\label{eq1.25} \sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)a_k \le \delta(\lambda p+\alpha-\gamma)\,, \end{align}
hence, (\ref{eq1.24}) is true if
\begin{align}\tag{26}\label{eq1.26} \frac{k\left|z\right|^{k-p}}{p-\rho}< \frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\,. \end{align}
Solving (\ref{eq1.26}) for \(\left|z\right|\), we obtain \begin{align*} \left|z\right|< \left\lbrace\frac{[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)(p-\rho)}{\delta(\lambda p+\alpha-\gamma)} \right\rbrace^{\frac{1}{k-p}}, & & (k\ge p+1).\end{align*} Hence, the proof.

Theorem 5. If \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\),then \(f(z)\) is p-valently starlike of order \(\rho\) in \(\left|z\right|< r_{2}(\lambda,\alpha,\gamma,\delta,\rho)\), where \begin{align*} r_{2}(\lambda,\alpha,\gamma,\delta,\rho)=\inf_{k}\left\lbrace\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)(p-\rho)}{\delta(\lambda p+\alpha-\gamma)(k-p)} \right\rbrace^{\frac{1}{k-p}},& & (k\ge p+1).\end{align*}

Proof Let \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then \[\left|\frac{zf'(z)}{f(z)}-p\right|< p-\rho\,.\] The inequality

\begin{align}\tag{27} \left|\frac{zf'(z)}{f(z)}-p\right|& =\left|\frac{z(pz^{p-1}-\sum_{k=p+1}^{\infty}ka_{k}z^{k-1})-p(z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k})}{z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k}}\right|\notag\\ \label{eq1.27}& =\left|\frac{pz^{p}-\sum_{k=p+1}^{\infty}ka_{k}z^{k}-pz^{p}+p\sum_{k=p+1}^{\infty}a_{k}z^{k})}{z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k}}\right|\,. \end{align}
Since
\begin{align}\tag{28}\label{eq1.28} \left|\frac{-\sum_{k=p+1}^{\infty}(k-p)ka_{k}z^{k}}{z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k}}\right|= \left|\frac{\sum_{k=p+1}^{\infty}(k-p)a_{k}z^{k-p}}{1-\sum_{k=p+1}^{\infty}a_{k}z^{k-p}}\right|\le \frac{\sum_{k=p+1}^{\infty}(k-p)a_{k}\left| z\right| ^{k-p}}{1-\sum_{k=p+1}^{\infty}a_{k}\left|z\right|^{k-p}}< p-\rho\,, \end{align}
i.e., \begin{align*} \sum_{k=p+1}^{\infty}ka_{k}\left| z\right| ^{k-p}-\sum_{k=p+1}^{\infty}pa_{k}\left| z\right| ^{k-p}&< (p-\rho)(1-\sum_{k=p+1}^{\infty}a_{k}\left|z\right|^{k-p})\\ & = \sum_{k=p+1}^{\infty}ka_{k}\left| z\right| ^{k-p}-\sum_{k=p+1}^{\infty}pa_{k}\left| z\right| ^{k-p}\end{align*} \begin{align*} & < p-\sum_{k=p+1}^{\infty}pa_{k}\left|z\right|^{k-p}-\rho+\sum_{k=p+1}^{\infty}\rho a_{k}\left|z\right|^{k-p}\\ & = \sum_{k=p+1}^{\infty}ka_{k}\left|z\right|^{k-p}-\sum_{k=p+1}^{\infty}\rho a_{k}\left|z\right|^{k-p}\\ &< p-\rho\\ & = \sum_{k=p+1}^{\infty}\frac {(k-\rho)a_{k}\left|z\right|^{k-p}}{p-\rho}\\& < 1.\end{align*} Since \[ \sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)a_k \le \delta(\lambda p+\alpha-\gamma)\,. \] This holds true if \[\frac {(k-\rho)\left|z\right|^{k-p}}{p-\rho}< \frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\,,\] \begin{align*} \left|z\right|< \left\lbrace\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)(p-\rho)}{(k-p)\delta(\lambda p+\alpha-\gamma)} \right\rbrace^{\frac{1}{k-p}} , & & (k\ge p+1),\end{align*} hence, the proof.

Theorem 6. If \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then \(f(z)\) is p-valently convex of order \(\rho\) in \(\left|z\right|< r_{3}(\lambda,\alpha,\gamma,\delta,\rho)\), where \begin{align*} r_{3}(\lambda,\alpha,\gamma,\delta,\rho)=\inf_{k}\left\lbrace\frac{[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)p(p-\rho)}{\delta(\lambda p+\alpha-\gamma)(k-p)} \right\rbrace^{\frac{1}{k-p}}, & & (k\ge p+1).\end{align*}

Proof Let \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then \[\left|1+\frac{zf”(z)}{f'(z)}-p\right|< p-\rho\,.\] The inequality \begin{align*} & \left|1+\frac{zf''(z)}{f'(z)}-p\right|\\ & =\left|\frac{pz^{p-1}-\sum_{k=p+1}^{\infty}ka_{k}z^{k-1}+z(p(p-1)z^{p-2}-\sum_{k=p+1}^{\infty}k(k-1)a_{k}z^{k-2})-p(pz^{p-1}-\sum_{k=p+1}^{\infty}ka_{k}z^{k-1})}{pz^{p-1}-\sum_{k=p+1}^{\infty}ka_{k}z^{k-1}}\right|\\ & =\left|\frac{pz^{p-1}-\sum_{k=p+1}^{\infty}ka_{k}z^{k-1}+p(p-1)z^{p-1}-\sum_{k=p+1}^{\infty}k(k-1)a_{k}z^{k-1})-p^{2}z^{p-1}+\sum_{k=p+1}^{\infty}pka_{k}z^{k-1})}{pz^{p-1}-\sum_{k=p+1}^{\infty}ka_{k}z^{k-1}}\right|\\ & =\left|\frac{-\sum_{k=p+1}^{\infty}k(k-p)a_{k}z ^{k-p}}{p-\sum_{k=p+1}^{\infty}ka_{k}z^{k-p}}\right|\\ & \le \frac{\sum_{k=p+1}^{\infty}k(k-p)a_{k}\left| z\right| ^{k-p}}{p-\sum_{k=p+1}^{\infty}ka_{k}\left|z\right|^{k-p}}\\ & < p-\rho.\end{align*} So, \[\sum_{k=p+1}^{\infty}k(k-p)a_{k}\left| z\right| ^{k-p}< (p-\rho)(p-\sum_{k=p+1}^{\infty}ka_{k}\left|z\right|^{k-p})\,,\] implies \[\sum_{k=p+1}^{\infty}k(k-p)a_{k}\left| z\right| ^{k-p}< p(p-\rho)-(p-\rho)\sum_{k=p+1}^{\infty}ka_{k}\left|z\right|^{k-p})\,,\] implies \[\sum_{k=p+1}^{\infty}k(k-\rho)a_{k}\left| z\right|^{k-p}< p(p-\rho)\,.\] Since \( \sum\limits_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)a_k \le \delta(\lambda p+\alpha-\gamma) \). This is true if \[\frac{k(k-\rho)\left| z\right|^{k-p}}{p(p-\rho)}< \frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\,,\] \begin{align*} \left|z\right|< \left\lbrace\frac{[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)p(p-\rho)}{(k-p)\delta(\lambda p+\alpha-\gamma)} \right\rbrace^{\frac{1}{k-p}},& & (k\ge p+1),\end{align*} hence, the proof.

Theorem 7. Let

\begin{align}\tag{29}\label{eq1.29} f_{p}(z)=z^{p} , f_{k}(z)=z^{p}-\frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}z^{k},& & (k\ge p+1), \end{align}
then, \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) if and only if it can be expressed in the form \[f(z)=\lambda_{p}f_{p}(z)+\sum_{k=p+1}^{\infty}\lambda_{k}f_{k}(z)\,,\] where \(\lambda_{p}\ge 0\) and \(\lambda_{p}\)=\(1 – \sum\limits_{k=p+1}^{\infty}\lambda_{k}\).

Proof Assume that \(f(z)=\lambda_{p}f_{p}(z)+\sum_{k=p+1}^{\infty}\lambda_{k}f_{k}(z)\), then

\begin{align}\tag{30}\label{eq1.30} f(z)=(1 – \sum_{k=p+1}^{\infty}\lambda_{k})z^{p}+\sum_{k=p+1}^{\infty}\lambda_{k}\left\{ z^{p}-\frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}z^{k}\right\}\,, \end{align}
implies \[f(z)=z^{p}-\sum_{k=p+1}^{\infty}\lambda_{k}\left\{ \frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}\right\}z^{k}\,.\] Thus, \begin{align*} &\sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)\lambda_{k} \frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}\\ &= \delta(\lambda p+\alpha-\gamma) \sum_{k=p+1}^{\infty}\lambda_{k} = \delta(\lambda p+\alpha-\gamma)(1-\lambda_{p}) \le \delta(\lambda p+\alpha-\gamma)\,,\end{align*} which shows that \(f(z)\) satisfies condition (\ref{eq1.15}) and therefore, \(f\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta).\) Conversely, suppose that \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), since \begin{align*} a_{k} \le \frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)},& & (k\ge p+1),\end{align*} we may set \begin{align*} & \lambda_{k}=\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}a_{k},\\ & \lambda_{p}=1- \sum_{k=p+1}^{\infty}\lambda_{k}\,,\end{align*} then we obtain from \(f(z)= z^{p}-\sum_{k=p+1}^{\infty}a_{k}z^{k}\), \[ f(z)=(\lambda_{p}+\sum_{k=p+1}^{\infty}\lambda_{k})z^{p}- \sum_{k=p+1}^{\infty}\lambda_{k} \frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}z^{k}\,,\] i.e., \[ f(z)=\lambda_{p}z^{p}+ \sum_{k=p+1}^{\infty}\lambda_{k}(z^{p}- \frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}z^{k})\,,\] implies \[ f(z)=\lambda_{p}z^{p} + \sum_{k=p+1}^{\infty}\lambda_{k}f_{k}(z)\,.\] This completes the proof.

Corollary 2. The extreme points of \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) are given by; \begin{align*} & f_{p}(z)=z^{p}, &\\ &f_{k}(z)=z^{p} – \frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}z^{k} , & (k\ge p+1).\end{align*}

Theorem 8. Let \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) and suppose that

\begin{align}\tag{31}\label{eq1.31} \sum_{j=p+1}^{\infty}(j-q)_{q+1}a_{j} \le \frac{\delta(\lambda p+\alpha-\gamma) \Gamma (k+1)\Gamma (2+p-l-q)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)\Gamma (k+1-l-q)\Gamma (p+1-q)}\,, \end{align}
for some \(0\le q \le j\), \(0\le l< 1\), \((j-q)_{q+1}\) denotes the pochhammer symbol defined by \((j-q)_{q+1}=(j-q)(j-q+1)…j.\) Also, let the function
\begin{align}\tag{32}\label{eq1.32} f_{k}(z)=z^{p}-\frac{\delta(\lambda p+\alpha-\gamma)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}z^{k} (k\ge p+1)\,. \end{align}
If there exists an analytic function \(w(z)\) defined by
\begin{align}\tag{33}\label{eq1.33} (w(z))^{k-p}=\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\frac{\Gamma (k+1-l-q)}{\Gamma (k+1)}\sum_{j=p+1}^{\infty}(j-q)_{q+1\psi (j) a_{j}z^{j-p}}\,, \end{align}
with \((k\ge q)\) \begin{align*}\Psi (j) =\frac{\Gamma (j-q)}{\Gamma (j+1-l-q)},& & (0\le l0\) and \(z=re^{i\theta}\), \((0 < r< 1)\),
\begin{align}\tag{34}\label{eq1.34} \int_{0}^{2\pi}\left|D_{z}^{q+l}f(z)\right|^{\sigma}d\theta \le \int_{0}^{2\pi}\left|D_{z}^{q+l}f_{k}(z)\right|^{\sigma}d\theta\,. \end{align}

Proof Let \(f(z)=z^{p}-\sum_{j=p+1}^{\infty}a_{j}z^{j}.\) By means of (\ref{eq1.12}) and Definition 6, we have

\begin{align} D_{z}^{q+l}f(z)& =\frac{\Gamma (p+1)z^{p-l-q}}{\Gamma (p+1-l-q)}-\sum_{j=p+1}^{\infty}\frac{\Gamma (j+1)}{\Gamma (j+1-l-q)}a_{j}z^{j-l-q}\notag\\ &= \frac{\Gamma (p+1)z^{p-l-q}}{\Gamma (p+1-l-q)}[1-\sum_{j=p+1}^{\infty}\frac{\Gamma (j+1)\Gamma (p+1-l-q)}{\Gamma (p+1)\Gamma (j+1-l-q)}a_{j}z^{j-p}]\tag{35}\\ \label{eq1.35} &= \frac{\Gamma (p+1)z^{p-l-q}}{\Gamma (p+1-l-q)}[1-\sum_{j=p+1}^{\infty}\frac{\Gamma (p+1-l-q)}{\Gamma (p+1) }(j-q)_{q+1}\Psi(j)a_{j}z^{j-p}\tag{36}]\,, \end{align}
where \( \Psi(j)=\frac{\Gamma (j-q)}{\Gamma (j+1-l-q)},\;\;\;\;(0\le l< 1, j\ge p+1).\) Since \(\psi\) is a decreasing function of \(j\), we get \[ 0< \Psi(j) \le \Psi(p+1)=\frac{\Gamma (p+1-q)}{\Gamma (2+p-l-q)}\,.\] Similarly, from (\ref{eq1.32}), (\ref{eq1.14}), and Definition 6, we have
\begin{align} D_{z}^{q+l}f_{k}(z)&=\frac{\Gamma (p+1)z^{p-l-q}}{\Gamma (p+1-l-q)}-\frac{\delta(\lambda p+\alpha-\gamma)\Gamma (k+1)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)\Gamma (k+1-l-q)}z^{k-l-q}\notag\\ \label{eq1.36} &=\frac{\Gamma (p+1)z^{p-l-q}}{\Gamma (p+1-l-q)}[1-\frac{\delta(\lambda p+\alpha-\gamma)\Gamma(k+1)\Gamma (p+1-l-q)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)\Gamma (p+1)\Gamma (k+1-l-q)}z^{k-p}]\,\tag{37}. \end{align}
For some \(\sigma>0\) and \(z=re^{i\theta}\), \((0 < r< 1)\), we show that
\begin{align}\label{eq1.37} \int_{0}^{2\pi}&\left|1-\sum_{j=p+1}^{\infty}\frac{\Gamma (p+1-l-q)}{\Gamma (p+1)}(j-q)_{q+1}\psi(j)a_{j}z^{j-p}\right|^{\sigma}d(\theta)\notag\\ &\le \int_{0}^{2\pi}\left|1-\frac{\delta(\lambda p+\alpha-\gamma)\Gamma (k+1)\Gamma (p+1-l-q)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)\Gamma (p+1)\Gamma (k+1-l-q)}z^{k-p}\right|^{\sigma}d(\theta)\,,\tag{38} \end{align}
so, by applying Lemma 1, it is enough to show that
\begin{align}\label{eq1.38} 1-&\sum_{j=p+1}^{\infty}\frac{\Gamma (p+1-l-q)}{\Gamma (p+1)}(j-q)_{q+1}\psi(j)a_{j}z^{j-p} \notag\\ &\prec 1-\frac{\delta(\lambda p+\alpha-\gamma)\Gamma (k+1)\Gamma (p+1-l-q)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)\Gamma (p+1)\Gamma (k+1-l-q)}z^{k-p}\,\tag{39}. \end{align}
If the above subordination holds true, then we have an analytic function \(w(z)\) with \(w(0)=0\), \(|w(z)|< 1\), such that
\begin{align}\label{eq1.39} 1-&\sum_{j=p+1}^{\infty}\frac{\Gamma (p+1-l-q)}{\Gamma (p+1)}(j-q)_{q+1}\psi(j)a_{j}z^{j-p} \notag\\ &= 1-\frac{\delta(\lambda p+\alpha-\gamma)\Gamma (k+1)\Gamma (p+1-l-q)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)\Gamma (p+1)\Gamma (k+1-l-q)}w(z)^{k-p}\,\tag{40}. \end{align}
By the condition of the Theorem, we define the function \(w(z)\) by
\begin{align}\tag{41}\label{eq1.40} (w(z))^{k-p}=\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\frac{\Gamma(k+1-l-q)}{\Gamma (k+1)}\sum_{j=p+1}^{\infty}(j-q)_{q+1}\psi(j)a_{j}z^{j-p}\,, \end{align}
which readily yields \(w(0)=0\). For such a function \(w(z)\), we have
\begin{align}\tag{42} \notag\left|(w(z))\right|^{k-p}&\le \frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\frac{\Gamma(k+1-l-q)}{\Gamma (k+1)}\sum_{j=p+1}^{\infty}(j-q)_{q+1}\psi(j)a_{j}\left| z\right| ^{j-p} \\ \notag&\le \left| z\right| \frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\frac{\Gamma(k+1-l-q)}{\Gamma (k+1)}\psi(p+1)\sum_{j=p+1}^{\infty}(j-q)_{q+1}a_{j}\\ \notag&= \left| z\right| \frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\frac{\Gamma(k+1-l-q)\Gamma(p+1-q)}{\Gamma (k+1)\Gamma(2+p-l-q)}\sum_{j=p+1}^{\infty}(j-q)_{q+1}a_{j}\\ \label{eq1.41} &\le \left|z\right|< 1. \end{align}
By means of the hypothesis of the theorem, the result is proved.

As a special case \(q=0\), we have following results from Theorem 8.

Corollary 3. Let \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) and suppose that

\begin{align}\tag{43}\label{eq1.42} \sum_{j=p+1}^{\infty}ja_{j} \le \frac{\delta(\lambda p+\alpha-\gamma)\Gamma (k+1)\Gamma (2+p-l)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda) \Gamma (k+1-l)\Gamma (p+1)},& & (j\ge p+1)\,, \end{align}
if there exists an analytic function \(w(z)\) defined by
\begin{align}\tag{44}\label{eq1.43} (w(z))^{k-p}=\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\frac{\Gamma (k+1-l)}{\Gamma (k+1)}\sum_{j=p+1}^{\infty}{j} \Psi (j) a_{j}z^{j-p}\,, \end{align}
with \begin{align*}\psi(j)=\frac{\Gamma(j)}{\Gamma(j+1-l)},& & (0\le l0\) and \(z=re^{i\theta}\),\;\;\; \((0 < r < 1)\)
\begin{align}\tag{45}\label{eq1.44} \int_{0}^{2\pi}\left|D_{z}^{l}f(z)\right|^{\sigma}d\theta \le \int_{0}^{2\pi}\left|D_{z}^{l}f_{k}(z)\right|^{\sigma}d\theta\,. \end{align}

Letting \(q=1\), we have the following from Theorem 8.

Corollary 4. Let \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) and suppose that

\begin{align}\tag{46}\label{eq1.45} \sum_{j=p+1}^{\infty}j(j-1)a_{j} \le \frac{\delta(\lambda p+\alpha-\gamma)\Gamma (k+1)\Gamma (p+1-l)}{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)\Gamma (k-l)\Gamma (p)}, & &(j\ge p+1)\,, \end{align}
if there exists an analytic function \(w(z)\) define by
\begin{align}\tag{47}\label{eq1.46} (w(z))^{k-p}=\frac{k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)}{\delta(\lambda p+\alpha-\gamma)}\frac{\Gamma (k-l)}{\Gamma (k+1)}\sum_{j=p+1}^{\infty}j(j-1) \Psi(j) a_{j}z^{j-p} \,,\end{align}
with \begin{align*} \psi(j)=\frac{\Gamma(j-1)}{\Gamma(j-l)},& & (0\le l0\) and \(z=re^{i\theta}\), \((0 < r < 1)\)
\begin{align}\tag{48}\label{eq1.47} \int_{0}^{2\pi}\left|D_{z}^{1+l}f(z)\right|^{\sigma}d\theta \le \int_{0}^{2\pi}\left|D_{z}^{1+l}f_{k}(z)\right|^{\sigma}d\theta,& & (0\le l < 1). \end{align}

Theorem 9. If \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then we have \[\left|D_{z}^{-l}f(z)\right|\le\frac{\Gamma(p+1)}{\Gamma(p+l+1)}\left|z\right|^{p+l}\left[1+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+l+1)}\left|z\right|\right]\,,\] and

\begin{align}\tag{49}\label{eq1.48} \left|D_{z}^{-l}f(z)\right|\ge\frac{\Gamma(p+1)}{\Gamma(p+l+1)}\left|z\right|^{p+l}\left[1-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+l+1)}\left|z\right|\right]\,. \end{align}

Proof Suppose that \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), using Theorem 1, we find that \[ \sum_{k=p+1}^{\infty}k[1+(\frac{k}{p}+\beta-\mu-1)t]^{n}(1+\delta\lambda)a_k \le \delta(\lambda p+\alpha-\gamma)\,, \] implies

\begin{align}\tag{50}\label{eq1.49} \frac{(p+1)[p+(p(\beta-\mu)+1)t]^{n}(1+\delta\lambda)}{p^{n}}\sum_{k=p+1}^{\infty}a_k \le \delta(\lambda p+\alpha-\gamma)\,, \end{align}
i.e.,
\begin{align}\tag{51}\label{eq1.50} \sum_{k=p+1}^{\infty}a_k \le \frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{(p+1)[p+(p(\beta-\mu)+1)t]^{n}(1+\delta\lambda)}\,. \end{align}
From (\ref{eq1.7}), if \(f(z)=z^{p}-\sum\limits_{k=p+1}^{\infty}a_{k}z^{k}\), \[D_{z}^{-l}f(z)=\frac{\Gamma(p+1)}{\Gamma(p+l+1)}z^{p+l}-\sum_{k=p+1}^{\infty}\frac{\Gamma(k+1)}{\Gamma(k+l+1)}a_{k}z^{k+l}\,,\] implies
\begin{align}\tag{52}\label{eq1.51} \frac {\Gamma(p+l+1)}{\Gamma(p+1)}z^{-l}D_{z}^{-l}f(z)=z^{p}-\sum_{k=p+1}^{\infty}\frac{\Gamma(k+1)\Gamma(p+l+1)}{\Gamma(p+1)\Gamma(k+l+1)}a_{k}z^{k}= z^{p}-\sum_{k=p+1}^{\infty}\Psi(k)a_{k}z^{k}\,, \end{align}
where
\begin{align}\tag{53}\label{eq1.52} \Psi(k)=\frac{\Gamma(k+1)\Gamma(p+l+1)}{\Gamma(p+1)\Gamma(k+l+1)}\,. \end{align}
Clearly, \(\psi\) is a decreasing function of \(k\) and we get \[0< \Psi(k)\le\Psi(p+1)=\frac{p+1}{p+l+1}\,.\] Using (\ref{eq1.50}) and (\ref{eq1.52}), we obtain, \begin{align*} \left|\frac{\Gamma(p+l+1)}{\Gamma(p+1)}z^{-l}D_{z}^{-l}f(z)\right|&\le\left|z\right|^{p}+\psi(p+1)\left|z\right|^{p+1}\sum_{k=p+1}^{\infty}a_{k}\\ &\le \left|z\right|^{p}+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+l+1)}\left|z\right|^{p+1}\,,\end{align*} which is equivalent to assertion (\ref{eq1.48}) and \begin{align*} \left|\frac{\Gamma(p+l+1)}{\Gamma(p+1)}z^{-l}D_{z}^{-l}f(z)\right|&\ge\left|z\right|^{p}-\psi(p+1)\left|z\right|^{p+1}\sum_{k=p+1}^{\infty}a_{k}\\ &\ge \left|z\right|^{p}-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+l+1)}\left|z\right|^{p+1}\,,\end{align*} which completes the proof.

Theorem 10. If \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then we have \[\left|D_{z}^{l}f(z)\right|\le\frac{\Gamma(p+1)}{\Gamma(p-l+1)}\left|z\right|^{p-l}\left[1+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p-l+1)}\left|z\right|\right]\,,\] and

\begin{align}\tag{54}\label{eq1.53} \left|D_{z}^{l}f(z)\right|\ge\frac{\Gamma(p+1)}{\Gamma(p-l+1)}\left|z\right|^{p-l}\left[1-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p-l+1)}\left|z\right|\right]\,. \end{align}

Proof If \(f(z)=z^{p}-\sum\limits_{k=p+1}^{\infty}a_{k}z^{k}\), then \[D_{z}^{l}f(z)=\frac{\Gamma(p+1)}{\Gamma(p-l+1)}z^{p-l}-\sum_{k=p+1}^{\infty}\frac{\Gamma(k+1)}{\Gamma(k-l+1)}a_{k}z^{k-l}\,,\] implies

\begin{align}\tag{55}\label{eq1.54} \frac {\Gamma(p-l+1)}{\Gamma(p+1)}z^{l}D_{z}^{l}f(z)=z^{p}-\sum_{k=p+1}^{\infty}\frac{\Gamma(k+1)\Gamma(p-l+1)}{\Gamma(p+1)\Gamma(k-l+1)}a_{k}z^{k}= z^{p}-\sum_{k=p+1}^{\infty}\Psi(k)a_{k}z^{k}\,, \end{align}
where
\begin{align}\tag{56}\label{eq1.55} \Psi(k)=\frac{\Gamma(p-l+1)\Gamma(k+1)}{\Gamma(p+1)\Gamma(k-l+1)}\,. \end{align}
Clearly, \(\Psi\) is a decreasing function of \(k\) and we get \[0< \Psi(k)\le\Psi(p+1)=\frac{p+1}{p-l+1}\,.\] Using (\ref{eq1.50}) and (\ref{eq1.55}), we obtain, \begin{align*} \left|\frac{\Gamma(p-l+1)}{\Gamma(p+1)}z^{l}D_{z}^{l}f(z)\right|&\le\left|z\right|^{p}+\psi(p+1)\left|z\right|^{p+1}\sum_{k=p+1}^{\infty}a_{k}\\ &\le \left|z\right|^{p}+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p-l+1)}\left|z\right|^{p+1}\,,\end{align*} which is equivalent to assertion (\ref{eq1.53}). Similarly, \begin{align*} \left|\frac{\Gamma(p-l+1)}{\Gamma(p+1)}z^{l}D_{z}^{l}f(z)\right|&\ge\left|z\right|^{p}-\Psi(p+1)\left|z\right|^{p+1}\sum_{k=p+1}^{\infty}a_{k}\\ &\ge \left|z\right|^{p}-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p-l+1)}\left|z\right|^{p+1}\,, \end{align*} which completes the proof.

Corollary 5. If \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then we have

\begin{align}\tag{57}\label{eq1.56} \notag \left|z\right|^{p}-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+1)}\left|z\right|^{p+1}&\le\left|f(z)\right|\\ &\le \left|z\right|^{p}+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+1)}\left|z\right|^{p+1}\,. \end{align}

Proof From Definition 4, we have \[\lim_{l\to 0} D_{z}^{-l}f(z)=f(z)\,.\] Therefore, setting \(l=0\) in (\ref{eq1.48}), we obtain \[\left|D_{z}^{0}f(z)\right|\le\frac{\Gamma(p+1)}{\Gamma(p+0+1)}\left|z\right|^{p+0}\left[1+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+0+1)}\left|z\right|\right]\,,\] and \begin{align*}%\label{eq1.57} \left|D_{z}^{0}f(z)\right|\ge\frac{\Gamma(p+1)}{\Gamma(p+0+1)}\left|z\right|^{p+0}\left[1-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+0+1)}\left|z\right|\right]\,, \end{align*} i.e., \begin{align*}%\label{eq1.58} \left|z\right|^{p}-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+1)}\left|z\right|^{p+1}&\le\left|f(z)\right|\notag\\ &\le \left|z\right|^{p}+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p+1)}\left|z\right|^{p+1}\,, \end{align*} which is (57).

Corollary 6. If \(f(z)\in S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\), then we have

\begin{align}\tag{58}p\left|z\right|^{p-1}-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)}\left|z\right|^{}\le\left|f'(z)\right| \label{eq1.59} \le p\left|z\right|^{p-1}+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)}\left|z\right|^{p}\,. \end{align}

Proof From Definition 4, we have \[\lim_{l\to 1} D_{z}^{l}f(z)=f'(z)\,.\] Therefore, setting \(l=1\) in (\ref{eq1.53}), we obtain \[\left|D_{z}^{1}f(z)\right|\le\frac{\Gamma(p+1)}{\Gamma(p)}\left|z\right|^{p-1}\left[1+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p)}\left|z\right|\right]\,,\] and \begin{align*} \left|D_{z}^{1}f(z)\right|\ge\frac{\Gamma(p+1)}{\Gamma(p)}\left|z\right|^{p-1}\left[1-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)(p)}\left|z\right|\right]\,, \end{align*} i.e., \[ p\left|z\right|^{p-1}-\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)}\left|z\right|^{p}\le\left|f'(z)\right| \le p\left|z\right|^{p-1}+\frac{\delta(\lambda p+\alpha-\gamma)p^{n}}{[p+(p(\beta-\mu)+1)t]^{n}(1+\delta \lambda)}\left|z\right|^{p}\,,\] which is (\ref{eq1.59}).

Acknowledgments :

The authors acknowledge the management of the University of Ilorin for providing us with a suitable research laboratory and library to enable us carried out this research.

Conflicts of Interest:

”The author declares no conflict of interest.”

Data Availability:

All data required for this research is included within this paper.

Funding Information:

No funding is available for this research.

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