In the present paper, we study a differential inequality involving certain differential operator. As a special case of our main result, we obtained certain differential inequalities implying sufficient conditions for meromorphic starlike and meromorphic convex functions of certain order.
Let \(\Sigma_{p,n}\) denote the class of functions of the form \[f(z)=\frac{a_{-1}}{z^p}+\sum_{k=n}^\infty a_{k-p}z^{k-p}~(p,n\in \mathbb N=\{1,2,3,\ldots\}),\] which are analytic and \(p\)-valent in the punctured unit disc \( \mathbb E_0=\mathbb E\setminus\{0\},\) where \(\mathbb E = \{z\in\mathbb C:|z|\alpha,(\alpha\alpha,(\alpha\alpha,(0\leq\alpha\alpha,(0\leq\alpha< 1;z \in \mathbb E)\right\}.\] Note that \(\mathcal{MS}^*(\alpha)=\mathcal{MS}^*_0(1,\alpha)\) and \(\mathcal{MK}(\alpha)=\mathcal{MK}_0(1,\alpha).\) In the theory of meromorphic functions, there exists a variety of results for starlikeness and convexity of meromorphic functions, we state some of them below. Wang et al. [1] proved the following results;
Theorem 1. If \(f(z)\in \Sigma_p\) satisfies the following inequality \[\left|\frac{f(z)}{z f'(z)}\left(1+\frac{z f”(z)}{f'(z)}-\frac{z f'(z)}{f(z)}\right)\right|< \mu~\left(0< \mu< \frac{1}{p}\right),\] then \(f\in\mathcal{MS}_p^*\left(\frac{p}{1+p\mu}\right)\).
Theorem 2. If \(f(z)\in \Sigma_p\) satisfies the inequality \[\left|\frac{z f'(z)}{f(z)}-\frac{z f”(z)}{f'(z)}-1\right|< \delta~(0< \delta< 1),\] then \(f\in\mathcal{MS}_p^*(p(1-\delta))\).
Theorem 3. If \(f(z) \in \Sigma_p\) satisfies the following inequality \[\Re\left(\frac{zf'(z)}{f(z)}+\beta\frac{z^2 f”(z)}{f(z)}\right)< \beta\lambda\left(\lambda+\frac{1}{2}\right)+\frac{1}{2}p\beta-\lambda \hspace{0.5 cm} (\beta\geq 0, p-\frac{1}{2}\leq \lambda \leq p),\] then \(f\in\mathcal{MS}_p^*(\lambda).\)
Goswami et al. [2] proved the following results;
Theorem 4. If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\), satisfies the following inequality \[\left| [z^p f(z)]^{\frac{1}{\alpha-p}}\left(\frac{z f'(z)}{f(z)}+\alpha\right)+p-\alpha\right|< \frac{(n+1)(p-\alpha)}{\sqrt{(n+1)^2+1}}, z\in\mathbb E, \] for some real values of \(\alpha~(0\leq\alpha< p)\),then \(f\in\mathcal{MS}_{p,n}^*(\alpha).\)
Theorem 5.If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\) satisfies the following inequality \[\left|\frac{ \gamma [z^p f(z)]^\gamma}{z}\left(\frac{z f'(z)}{f(z)}+p\right)\right| \leq\frac{(n+1)}{2\sqrt{(n+1)^2+1}}, z\in\mathbb E, \] for \(\gamma\leq-\frac{1}{p}\),then \(f\in\mathcal{MS}_{p,n}^*\left(p+\frac{1}{\gamma}\right).\)
Theorem 6. If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\), satisfies the inequality \[\left| \left(\frac{z^{p+1} f'(z)}{-p}\right)^{\frac{1}{\alpha-p}}\left(1+\frac{z f”(z)}{f'(z)}+\alpha\right)+p-\alpha\right|< \frac{(n+1)(p-\alpha)}{\sqrt{(n+1)^2+1}}, ~z\in\mathbb E ,\] for some real values of \(\alpha~(0\leq\alpha< p)\),then \(f\in\mathcal{MK}_{p,n}(\alpha).\)
Theorem 7. If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\), satisfies the inequality \[\left| \frac{1}{z} \left(\frac{z^{p+1} f'(z)}{-p}\right)^{\frac{1}{\alpha-p}}\left(1+\frac{z f”(z)}{f'(z)}+p\right)\right|\leq\frac{(n+1)(p-\alpha)}{2\sqrt{(n+1)^2+1}}, ~z\in\mathbb E ,\] for some real values of \(\alpha~(0\leq\alpha< p)\),then \(f\in\mathcal{MK}_{p,n}(\alpha).\)
From above stated results, we notice that a number of sufficient conditions for meromorphic starlike and meromorphic convex functions have been obtained in terms of differential inequalities in the literature of meromorphic functions. The study of such results is a source of motivation for us to produce the present paper. In the present paper, we study differential inequalities involving a differential operator. As particular cases of our main results, we derive certain sufficient conditions for meromorphic starlike and meromorphic convex functions.
Lemma 1. Suppose w is a nonconstant analytic function in \(\mathbb E\) with \(w(0)=0\). If \(|w(z)|\) attains its maximum value at a point \(z_0 \in\mathbb E\) on the circle \(|z|=r< 1,\) then \(z_0 w'(z_0)=m w(z_0),\) where \(m\geq1\), is some real number.
Theorem 8. Let \(f(z)\in\Sigma_p\) satisfy
0,\) then \(f(z)\in\mathcal {MS}_n^*(p,\alpha)\), where \(n\in\mathbb N_0=\mathbb N\cup\{0\} \) and \begin{eqnarray} M(p,\alpha,\beta,\gamma)& = &\begin{cases}\left(1-\frac{\alpha}{p}\right)^\gamma\left(\frac{1}{2}-\frac{\alpha}{p}\right)^{\beta}, 0\leq\alpha< \frac{p}{2},\nonumber \\ \left(1-\frac{\alpha}{p}\right)^{\gamma+\beta}\left(\frac{2}{2-\frac{\alpha}{p}}\right)^{\beta},~\frac{p}{2}\leq\alpha< p. \end{cases} \end{eqnarray}
Proof. We consider the following two cases separately.
Case (i). When \( 0\leq\alpha< \frac{p}{2}\). Writing \(\frac{\alpha}{p}=\mu\), we see that \(0\leq\mu< \frac{1}{2}.\) Define a function \(w\) as
Case (ii). When \(\frac{p}{2}\leq\alpha< p\), therefore we must have \(\frac{1}{2}\leq\mu< 1\), where \(\mu=\frac{\alpha}{p}.\) Let \(w\) be defined by
Theorem 9. Let \(f(z)\in\Sigma_p\) satisfy
0,\) then \(f(z)\in\mathcal {MK}_n(p,\alpha)\), where \(n\in\mathbb N_0=\mathbb N\cup\{0\}\) and \begin{eqnarray} M(p,\alpha,\beta,\gamma)& = &\begin{cases}\left(1-\frac{\alpha}{p}\right)^\gamma\left(\frac{1}{2}-\frac{\alpha}{p}\right)^{\beta}, 0\leq\alpha< \frac{p}{2},\nonumber \\ \left(1-\frac{\alpha}{p}\right)^{\gamma+\beta}\left(\frac{2}{2-\frac{\alpha}{p}}\right)^{\beta},~\frac{p}{2}\leq\alpha< p. \end{cases} \end{eqnarray}
Proof. Again,we consider the following two cases separately.
Case (i). When \( 0\leq\alpha< \frac{p}{2}\). Writing \( \frac{\alpha}{p}=\mu\), we see that \(0\leq\mu< \frac{1}{2}.\) Define a function \(w\) as
Corollary 1. Let \(f\in\Sigma_p\) satisfy the condition \begin{eqnarray} \left|\frac{1}{p}\left(\frac{z f'(z)}{ f(z)}\right)-1\right|^\gamma \left|\frac{1}{p}\left(\frac{z f”(z)+3f'(z)}{z f'(z)+2f(z)}\right)-1\right|^\beta &\hspace{-0.3cm}<&\hspace{-0.3cm}\begin{cases}\left(1-\frac{\alpha}{p}\right)^\gamma \left(\frac{1}{2}-\frac{\alpha}{p}\right)^\beta, ~0\leq\alpha< \frac{p}{2},\nonumber \\ \left(1-\frac{\alpha}{p}\right)^{\gamma+\beta} \left(\frac{2}{2-\frac{\alpha}{p}}\right)^\beta,~\frac{p}{2}\leq\alpha< p, \end{cases} \end{eqnarray} for all \(z\in\mathbb E\) and for some real numbers \(\alpha(0\leq\alpha0\), then \(f\in\mathcal{MS}^*(p,\alpha)\).
For \(p=1\), Theorem 8 reduces to the following;
Corollary 2. For some real numbers \(\alpha(0\leq\alpha0\), if \(f\in\Sigma\) satisfies \begin{eqnarray} \left|\frac{D^{n+1}[f](z)}{D^n[f](z)}-1\right|^\gamma \left|\frac{D^{n+2}[f](z)}{D^{n+1}[f](z)}-1\right|^\beta &\hspace{-0.3cm}<&\hspace{-0.3cm}\begin{cases}\left(1-\alpha\right)^\gamma \left(\frac{1}{2}-\alpha\right)^\beta, ~0\leq\alpha< \frac{1}{2},\nonumber \\ \left(1-\alpha\right)^{\gamma+\beta} \left(\frac{2}{2-\alpha}\right)^\beta,~\frac{1}{2}\leq\alpha< 1, \end{cases} \end{eqnarray} in \(\mathbb E\), then \(\mathcal{MS}^*_n(1,\alpha)\), where \(n\in\mathbb {N}_0.\)
Setting \(n=0\) in above corollary, yields the following result.
Corollary 3. Let \(f(z)\in\Sigma\) satisfy the condition \begin{eqnarray} \left|\frac{z f'(z)}{f(z)}+1\right|^\gamma\left|\frac{z f”(z)+3f'(z)}{z f'(z)+2f(z)}-1\right|^\beta & < &\begin{cases}\left(1-\alpha\right)^\gamma \left(\frac{1}{2}-\alpha\right)^\beta, ~0\leq\alpha< \frac{1}{2},\nonumber \\ \left(1-\alpha\right)^{\gamma+\beta} \left(\frac{2}{2-\alpha}\right)^\beta,~\frac{1}{2}\leq\alpha< 1, \end{cases} \end{eqnarray} where \(z\in\mathbb E, \alpha~(0\leq\alpha0\),then \(f\in\mathcal{MS}^*(\alpha)\).
Setting \(\beta=\gamma=1\) and \(\alpha=0\) in above corollary, we obtain the following result.
Remark 1. If \(f(z)\in\Sigma\) satisfies \[\left|\frac{z f'(z)}{f(z)}+1\right|\left|\frac{z f”(z)+3f'(z)}{z f'(z)+2f(z)}-1\right|< \frac{1}{2},~z\in\mathbb E,\] then \(f\in\mathcal{MS}^*\).
By writing \(\beta=1\) and \(\gamma=0\), Theorem 8, we get
Corollary 4. If for all \(z\in\mathbb E\), a function \(f\in\Sigma_p\) satisfies the condition \begin{eqnarray} \frac{D^{n+2}[f](z)}{D^{n+1}[f](z)}&\hspace{-0.3cm}\prec&\begin{cases} 1+\left(\frac{1}{2}-\frac{\alpha}{p}\right)z ,~0\leq\alpha< \frac{p}{2},\nonumber \\ 1+\left[\frac{2\left(1- \frac{\alpha}{p}\right)}{2-\frac{\alpha}{p}}\right]z,~\frac{p}{2}\leq\alpha
\frac{\alpha}{p}.\]
Setting \(n=0\) in Theorem 9, we obtain the following result.
Corollary 5. Let \(f\in\Sigma_p\) satisfy the condition \begin{eqnarray} \left|\frac{1}{p}\left(\frac{z f”(z)+3f'(z)}{ f'(z)}\right)-1\right|^\gamma \left|\frac{1}{p}\left(\frac{z^2 f”'(z)+7zf'(z)+9f'(z)}{z f”(z)+3f(z)}\right)-1\right|^\beta &<&\hspace{-0.3cm}\begin{cases}\left(1-\frac{\alpha}{p}\right)^\gamma \left(\frac{1}{2}-\frac{\alpha}{p}\right)^\beta, ~0\leq\alpha< \frac{p}{2},\nonumber \\ \left(1-\frac{\alpha}{p}\right)^{\gamma+\beta} \left(\frac{2}{2-\frac{\alpha}{p}}\right)^\beta,~\frac{p}{2}\leq\alpha< p, \end{cases} \end{eqnarray} for all \(z\in\mathbb E\) and for some real numbers \(\alpha(0\leq\alpha0,\) then \(f\in\mathcal{MK}(p,\alpha)\).
For \(p=1\), Theorem 9 reduces to the following;
Corollary 6. For some real numbers \(\alpha(0\leq\alpha0\), if \(f\in\Sigma\) satisfies \begin{eqnarray} \left|\frac{(D^{n+1}[f](z))’}{(D^n[f](z))’}-1\right|^\gamma \left|\frac{(D^{n+2}[f](z))’}{(D^{n+1}[f](z))’}-1\right|^\beta &<&\begin{cases}\left(1-\alpha\right)^\gamma \left(\frac{1}{2}-\alpha\right)^\beta, ~0\leq\alpha< \frac{1}{2},\nonumber \\ \left(1-\alpha\right)^{\gamma+\beta} \left(\frac{2}{2-\alpha}\right)^\beta,~\frac{1}{2}\leq\alpha< 1, \end{cases} \end{eqnarray} in \(\mathbb E\),then \(\mathcal{MK}_n(1,\alpha)\), where \(n\in\mathbb {N}_0.\)
Setting \(n=0\) in above corollary, yields the following result;
Corollary 7. Let \(f(z)\in\Sigma\) satisfy the condition \begin{eqnarray} \left|\frac{z f”(z)}{f'(z)}+2\right|^\gamma\left|\frac{z^2 f”'(z)+7zf”(z)+9f'(z)}{z f”(z)+3f(z)}-1\right|^\beta & < &\begin{cases}\left(1-\alpha\right)^\gamma \left(\frac{1}{2}-\alpha\right)^\beta, ~0\leq\alpha< \frac{1}{2},\nonumber \\ \left(1-\alpha\right)^{\gamma+\beta} \left(\frac{2}{2-\alpha}\right)^\beta,~\frac{1}{2}\leq\alpha< 1, \end{cases} \end{eqnarray} where \(z\in\mathbb E, \alpha~(0\leq\alpha0\),then \(f\in\mathcal{MK}(\alpha)\).
Setting \(\beta=\gamma=1\) and \(\alpha=0\) in above corollary, we obtain the following result;
Remark 2. If \(f(z)\in\Sigma\) satisfies \[\left|\frac{z f”(z)}{f'(z)}+2\right|\left|\frac{z^2 f”'(z)+7zf”(z)+9f'(z)}{z f”(z)+3f(z)}-1\right|< \frac{1}{2},~z\in\mathbb E,\] then \(f\in\mathcal{MK}\).