A differential inequality and meromorphic starlike and convex functions

1. Introduction

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 \(\mathbb E = \{z\in\mathbb C:|z|< 1\}\). Define \begin{eqnarray*} D^0f(z)&=&f(z),\\ D^1 f(z)&=&\frac{a_{-1}}{z^p}+2a_0+3a_1z+4a_2z^2+\ldots=\frac{(z^2 f(z))'}{z},\\ D^2f(z)&=&D^1(D^1 f(z)), \end{eqnarray*} and for \(n=1,2,3,\ldots\) \[D^nf(z)=D^1(D^{n-1} f(z))=\frac{(z^2 D^{n-1}f(z))'}{z}.\] Let \(\mathcal {MS}^*_n(p,\alpha)\) denote the class of functions \(f \in \Sigma_{p,n} \) if \[-\Re\frac{1}{p}\left(\frac{D^{n+1}f(z)}{D^nf(z)}-2\right)>\alpha,(\alpha< 1;z \in \mathbb E).\] and let \(\mathcal {MK}_n(p,\alpha)\) denote the class of functions \(f \in \Sigma_{p,n} \) if \[-\Re\frac{1}{p}\left(\frac{(D^{n+1}f(z))'}{(D^nf(z))'}-2\right)>\alpha,(\alpha< 1;z \in \mathbb E).\] The classes of meromorphic starlike functions of order \(\alpha\) and meromorphic convex functions of order \(\alpha\) are denoted by \(\mathcal{MS}^*(\alpha)\) and \(\mathcal{MK}(\alpha)\), respectively and are defined as: \[ \mathcal{MS}^*(\alpha)=\left\{f\in\Sigma:-\Re\left(\frac{zf'(z)}{f(z)}\right)>\alpha,(0\leq\alpha< 1;z \in \mathbb E)\right\},\] and \[\mathcal{MK}(\alpha)=\left\{f\in\Sigma:-\Re\left(1+\frac{zf''(z)}{f'(z)}\right)>\alpha,(0\leq\alpha< 1;z \in \mathbb E)\right\}.\] 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.

2. Preliminaries

We shall use the following lemma of [3] to prove our result.

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.

3. Main Results

Theorem 8. Let \(f(z)\in\Sigma_p\) satisfy

for some real numbers \(\alpha, \beta\) and \(\gamma\) such that \(0\leq\alpha< p\), \(\beta\geq0\), \(\gamma\geq0,\beta+\gamma>0,\) 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

where \(w\) is an analytic function in \(\mathbb E, ~w(0)=0\) and \(w(z)\neq 1\) in \(\mathbb E\). Differentiating (2) logarithmatically, we get From the definition, we have the identity Using (4) in (3), we get \(\frac{D^{n+2}[f](z)}{D^{n+1}[f](z)}=\frac{1+(2\mu-3)w(z)}{1-w(z)}+\frac{2(\mu-1)zw'(z)}{(1-w(z))(1+(2\mu-3)w(z))}\). So, we have We claim that \(|w(z)|< 1,~z\in\mathbb E.\) Suppose, to the contrary, that there exists a point \(z_0\in\mathbb E\) such that \(\max_{|z|\leq|z_0|} |w(z)|=|w(z_0)|=1\). Then by Lemma 1, we have \(w(z_0)=e^{i\theta},0\leq\theta< 2\pi\) and \(z_0w'(z_0)=mw(z_0),m\geq1.\) Therefore \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&=&\left|\frac{2(1-\mu)w(z_0)}{1-w(z_0)}\right|^{\gamma+\beta}\left|1+ \frac{m}{1+(2\mu-3)w(z_0)}\right|^\beta \nonumber\\ &=&\frac{2^{\gamma+\beta}(1-\mu)^{\gamma+\beta}}{|1-e^{i\theta}|^{\gamma+\beta}} \left|1+ \frac{m}{1+(2\mu-3)e^{i\theta}}\right|^\beta \nonumber\\ &\geq&(1-\mu)^{\gamma+\beta}\left(1- \frac{m}{2(1-\mu)}\right)^\beta \nonumber\\ &\geq&(1-\mu)^{\gamma+\beta}\left(1- \frac{1}{2(1-\mu)}\right)^\beta \nonumber\\ &=&(1-\mu)^\gamma \left(\frac{1}{2}-\mu\right)^\beta,\nonumber \end{eqnarray} which contradicts (1) for \( 0\leq\alpha< \frac{p}{2}.\) Therefore we must have \(|w(z)|< 1\) for all \(z\in\mathbb E,\) and hence from (2), we conclude that \(f\in\mathcal {MS}_n^*(p,\alpha)\).

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

where \( w(z)\neq\frac{\mu}{1-\mu}\) in \(\mathbb E\). Then w is analytic in \(\mathbb E\) with \(w(0)=0\). Proceeding as in Case (i) above, we obtain We shall prove that \(|w(z)|< 1,~z \in\mathbb E.\) If not, suppose there exists a point \(z_0\in\mathbb E\) such that there exists a point \(z_0\in\mathbb E\) such that \(\max_{|z|\leq|z_0|} |w(z)|=|w(z_0)|=1\). Then by Lemma 1 we have \(w(z_0)=e^{i\theta},0\leq\theta< 2\pi\) and \( z_0w'(z_0)=mw(z_0),m\geq1.\) Therefore \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&=&\frac{(1-\mu)^{\gamma+\beta}\left(1+\frac{m\mu}{\mu+2-2\mu}\right)^\beta}{|\mu-(1-\mu)w(z_0)|^{\gamma+\beta}}\nonumber\\ &\geq& \left(1-\mu\right)^{\gamma+\beta}\left(\frac{2}{2-\mu}\right)^\beta,\nonumber \end{eqnarray*} which contradicts (1) for \(\frac{p}{2}\leq\alpha< p.\) Therefore, we must have \(|w(z)|< 1\) for all \(z\in\mathbb E\), and hence in view of (6), we conclude that \(f\in\mathcal {MS}_n^*(p,\alpha)\). This completes the proof of the theorem.

Theorem 9. Let \(f(z)\in\Sigma_p\) satisfy

for some real numbers \(\alpha, \beta\) and \(\gamma\) such that \(0\leq\alpha< p\), \(\beta\geq0\), \(\gamma\geq0,\beta+\gamma>0,\) 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

where \(w\) is an analytic function in \(\mathbb E, ~w(0)=0\) and \(w(z)\neq 1\) in \(\mathbb E\). Differentiating (9) logarithmatically, we get From the following identity we have Using the identity (12), Equation (10) may be written as \[\frac{(D^{n+2}[f](z))'}{(D^{n+1}[f](z))'}=\frac{1+(2\mu-3)w(z)}{1-w(z)}+\frac{2(\mu-1)zw'(z)}{(1-w(z))(1+(2\mu-3)w(z))}.\] So, we have The remaining part of the proof is similar to that of Theorem 8.

4. Criteria for Starlikeness and Convexity

When we assign particular values to various parameters involved in Theorem 8 and Theorem 9, we obtain following special cases. Setting \(n=0\) in Theorem 8, we obtain the following result.

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 \(\alpha(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\), then \(f\in\mathcal{MS}^*(p,\alpha)\).

For \(p=1\), Theorem 8 reduces to the following;

Corollary 2. For some real numbers \(\alpha(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\), 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 \(z\in\mathbb E, \alpha~(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\),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< p, \end{cases} \end{eqnarray} then \[2-\frac{D^{n+1}[f](z)}{D^n[f](z)}\prec\frac{1+\left(1-\frac{2\alpha}{p}\right)z}{1-z},~z\in\mathbb E, \] i.e. \[\Re\left(2-\frac{D^{n+1}[f](z)}{D^n[f](z)}\right)>\frac{\alpha}{p}.\]

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 \(\alpha(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0,\) then \(f\in\mathcal{MK}(p,\alpha)\).

For \(p=1\), Theorem 9 reduces to the following;

Corollary 6. For some real numbers \(\alpha(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\), 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 \(z\in\mathbb E, \alpha~(0\leq\alpha< 1), \beta\geq0\) and \(\gamma\geq0\) with \(\beta+\gamma>0\),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 \(f\in\mathcal{MK}\).

Acknowledgments

The authors are grateful to the referee for careful reading of the paper and valuable suggestions and comments.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflict of Interests

The authors declare no conflict of interest.