On a certain new subclass of meromorphic close-to-convex functions

Abstract

In this paper, we introduce and investigate a new subclass $M{K}^{(k)}(\beta ,\gamma )$ of meromorphic close-to-convex functions. For functions belonging to the class $M{K}^{(k)}(\beta ,\gamma )$, we obtain some coefficient inequalities and a distortion theorem. The results presented here would unify and extend some recent work of Wang et al. (Appl. Math. Lett. 25:454-460, 2012).

MSC:30C45.

1 Introduction

Let Σ be the class of functions f of the form:

$$f(z)=\frac{1}{z}+\sum _{n=1}^{\mathrm{\infty }}{a}_n{z}^n,$$

(1.1)

which are analytic in the punctured open unit disk $U^{\ast }=\{z\in C:0<|z|<1\}=U\setminus \{0\}$.

Let P denote the class of functions p given by

$$p(z)=1+\sum _{n=1}^{\mathrm{\infty }}{p}_n{z}^n(z\in U),$$

(1.2)

which are analytic and convex in U and satisfy the condition $\mathrm{\Re }(p(z))>0$ ($z\in U$).

A function $f\in \mathrm{\Sigma }$ is said to be in the class $M{S}^{\ast }(\alpha )$ of meromorphic starlike functions of order α if it satisfies the inequality

$$\mathrm{\Re }\left(\frac{z{f}^{\prime }(z)}{f(z)}\right)<-\alpha (z\in U^{\ast };0\le \alpha <1).$$

In addition, a function $f\in \mathrm{\Sigma }$ is said to be in the class MC of meromorphic close-to-convex functions if it satisfies the inequality

$$\mathrm{\Re }\left(\frac{z{f}^{\prime }(z)}{g(z)}\right)<0(z\in U^{\ast };g\in M{S}^{\ast }(0)=M{S}^{\ast }).$$

Recently, Srivastava et al. [1] (see also [2, 3]) introduced and studied the class $M{S}_s^{\ast }$ of meromorphic starlike functions with respect to symmetric points, which satisfies the condition

$$\mathrm{\Re }\left(\frac{z{f}^{\prime }(z)}{f(z)-f(-z)}\right)<0(z\in U^{\ast }).$$

More recently, Wang et al. [4] discussed a class MK of meromorphic close-to-convex functions, that is, a function $f\in \mathrm{\Sigma }$ is said to be in the class MK if it satisfies the inequality

$$\mathrm{\Re }\left(\frac{f^{\prime }(z)}{g(z)g(-z)}\right)>0(z\in U^{\ast }),$$

where $g\in M{S}^{\ast }(\frac{1}{2})$.

Let $f(z)=z+a_2{z}^2+\cdots$ be analytic in U. If there exists a function $g\in S^{\ast }(\frac{1}{2})$, such that

$$|\frac{z^2{f}^{\prime }(z)}{g(z)g(-z)}+1|<|\frac{z^2{f}^{\prime }(z)}{g(z)g(-z)}-1+2\gamma |(z\in U),$$

then we say that $f\in K_s(\gamma )$, $0\le \gamma <1$, where $S^{\ast }(\frac{1}{2})$ denotes the usual class of starlike functions of order 1/2. The function class $K_s(\gamma )$ was introduced and studied recently by Kowalczyk and Les-Bomba [5] (see also [6–9]).

For two functions f and g analytic in U, we say that the function $f(z)$ is subordinate to $g(z)$ in U, and we write $f(z)\prec g(z)$ ($z\in U$) if there exists a Schwarz function $w(z)$, analytic in U with $w(0)=0$ and $|w(z)|\le 1$, such that $f(z)=g(w(z))$ ($z\in U$). In particular, if the function g is univalent in U, then we have $f(0)=g(0)$ and $f(U)\subset g(U)$ (see, for example, [10]).

Motivated essentially by the above mentioned function classes MK and $K_s(\gamma )$, we now introduce a new class $M{K}^{(k)}(\beta ,\gamma )$ of meromorphic functions.

Definition 1 Let $M{K}^{(k)}(\beta ,\gamma )$ denote the class of functions in Σ satisfying the inequality

$$|\frac{z^{2-k}{f}^{\prime }(z)}{g_k(z)}+1|<\beta |\frac{z^{2-k}{f}^{\prime }(z)}{g_k(z)}+2\gamma -1|(z\in U^{\ast };0<\beta \le 1;0\le \gamma <1),$$

(1.3)

where $g\in M{S}^{\ast }(\frac{k-1}{k})$, $k\ge 1$, is a fixed positive integer and $g_k(z)$ is defined by the following equality:

$$g_k(z)=\prod _{v=0}^{k-1}{\epsilon }^{-v}g\left({\epsilon }^{v}z\right)({\epsilon }^k=1).$$

(1.4)

We note that $M{K}^{(2)}(1,0)=MK$ (see [4]), so the class $M{K}^{(k)}(\beta ,\gamma )$ is a generation of the class MK.

In this paper, we prove that the class $M{K}^{(k)}(\beta ,\gamma )$ is a subclass of meromorphic close-to-convex functions. Moreover, we provide some coefficient inequalities and a distortion theorem for functions in the class $M{K}^{(k)}(\beta ,\gamma )$. Our results unify and extend the corresponding results obtained by Wang et al. [4].

2 Main results

First of all, we give two meaningful conclusions about the class $M{K}^{(k)}(\beta ,\gamma )$. The proof of Theorem 1 below is much akin to that of Theorem 1 in [11], so we choose to omit the details involved.

Theorem 1 A function $f\in M{K}^{(k)}(\beta ,\gamma )$ if and only if there exists $g\in M{S}^{\ast }(\frac{k-1}{k})$ such that

$$-\frac{z^{2-k}{f}^{\prime }(z)}{g_k(z)}\prec \frac{1+(1-2\gamma )\beta z}{1-\beta z}(z\in U^{\ast }).$$

(2.1)

Remark 1

From Theorem 1, we know that

$$\mathrm{\Re }\left(\frac{z^{2-k}{f}^{\prime }(z)}{g_k(z)}\right)<0(z\in U^{\ast }),$$

(2.2)

because of $\mathrm{\Re }(\frac{1+(1-2\gamma )\beta z}{1-\beta z})>0$ ($z\in U^{\ast }$).

Lemma 1 Let ${\phi }_i\in M{S}^{\ast }({\alpha }_i)$, where $0\le {\alpha }_i<1$ ($i=0,1,\dots ,k-1$). Then for $k-1\le {\sum }_{i=0}^{k-1}{\alpha }_i<k$, we have

$$z^{k-1}\prod _{i=0}^{k-1}{\phi }_i(z)\in M{S}^{\ast }(\sum _{i=0}^{k-1}{\alpha }_i-(k-1)).$$

Proof Since ${\phi }_i\in M{S}^{\ast }({\alpha }_i)$ ($i=0,1,\dots ,k-1$), by the definition of meromorphic starlike functions, we have

$$\mathrm{\Re }\left(\frac{z{\phi }_0^{\mathrm{\prime }}(z)}{{\phi }_0(z)}\right)<-{\alpha }_0,\mathrm{\Re }\left(\frac{z{\phi }_1^{\mathrm{\prime }}(z)}{{\phi }_1(z)}\right)<-{\alpha }_1,\dots ,\mathrm{\Re }\left(\frac{z{\phi }_{k-1}^{\mathrm{\prime }}(z)}{{\phi }_{k-1}(z)}\right)<-{\alpha }_{k-1}.$$

(2.3)

We now let

$$F(z)=z^{k-1}{\phi }_0(z){\phi }_1(z)\cdots {\phi }_{k-1}(z).$$

(2.4)

Differentiating (2.4) with respect to z logarithmically, we easily get

$$\frac{z{F}^{\prime }(z)}{F(z)}=\frac{z{\phi }_0^{\mathrm{\prime }}(z)}{{\phi }_0(z)}+\frac{z{\phi }_1^{\mathrm{\prime }}(z)}{{\phi }_1(z)}+\cdots +\frac{z{\phi }_{k-1}^{\mathrm{\prime }}(z)}{{\phi }_{k-1}(z)}+(k-1).$$

(2.5)

From (2.5) together with (2.3), we obtain

$$\begin{array}{rcl}\mathrm{\Re }\left(\frac{z{F}^{\prime }(z)}{F(z)}\right)& =& \mathrm{\Re }\left(\frac{z{\phi }_0^{\mathrm{\prime }}(z)}{{\phi }_0(z)}\right)+\mathrm{\Re }\left(\frac{z{\phi }_1^{\mathrm{\prime }}(z)}{{\phi }_1(z)}\right)+\cdots +\mathrm{\Re }\left(\frac{z{\phi }_{k-1}^{\mathrm{\prime }}(z)}{{\phi }_{k-1}(z)}\right)+(k-1)\\ <& -\sum _{i=0}^{k-1}{\alpha }_i+(k-1)=-(\sum _{i=0}^{k-1}{\alpha }_i-(k-1))\end{array}$$

by noting that $0\le {\sum }_{i=0}^{k-1}{\alpha }_i-(k-1)<1$, which implies that

$$F(z)=z^{k-1}\prod _{i=0}^{k-1}{\phi }_i(z)\in M{S}^{\ast }(\sum _{i=0}^{k-1}{\alpha }_i-(k-1)).$$

The proof of Lemma 1 is thus completed. □

Theorem 2 Let $g(z)=\frac{1}{z}+{\sum }_{n=1}^{\mathrm{\infty }}{b}_n{z}^n\in M{S}^{\ast }(\frac{k-1}{k})$, then $z^{k-1}{g}_k(z)\in M{S}^{\ast }$.

Proof From (1.4), we know

$$\begin{array}{rcl}{z}^{k-1}{g}_k(z)& =& z^{k-1}\prod _{v=0}^{k-1}{\epsilon }^{-v}g\left({\epsilon }^{v}z\right)=z^{k-1}\prod _{v=0}^{k-1}{\epsilon }^{-v}[\frac{1}{{\epsilon }^{v}z}+\sum _{n=1}^{\mathrm{\infty }}{b}_n{\left({\epsilon }^{v}z\right)}^n]\\ =& z^{k-1}\prod _{v=0}^{k-1}[\frac{1}{{\epsilon }^{2v}z}+\sum _{n=1}^{\mathrm{\infty }}{b}_n{\epsilon }^{(n-1)v}{z}^n].\end{array}$$

(2.6)

Since $g(z)=\frac{1}{z}+{\sum }_{n=1}^{\mathrm{\infty }}{b}_n{z}^n\in M{S}^{\ast }(\frac{k-1}{k})$, by Lemma 1 and (2.6), we can easily get the assertion of Theorem 2. □

Remark 2 From Theorem 2 and the inequality (2.2), we see that if $f\in M{K}^{(k)}(\beta ,\gamma )$, then $f(z)$ is a meromorphic close-to-convex function. So, $M{K}^{(k)}(\beta ,\gamma )$ is a subclass of the class MC of meromorphic close-to-convex functions.

Next, we give some coefficient inequalities for functions belonging to the class $M{K}^{(k)}(\beta ,\gamma )$.

Theorem 3 Let $f(z)=\frac{1}{z}+{\sum }_{n=1}^{\mathrm{\infty }}{a}_n{z}^n$ and $g(z)=\frac{1}{z}+{\sum }_{n=1}^{\mathrm{\infty }}{b}_n{z}^n$ be analytic in $U^{\ast }$. If for $0<\beta \le 1$ and $0\le \gamma <1$, we have

$$\sum _{n=1}^{\mathrm{\infty }}n(1+\beta )|a_n|+\sum _{n=1}^{\mathrm{\infty }}(\beta |1-2\gamma |+1)|B_n|\le 2\beta (1-\gamma ),$$

(2.7)

where the coefficients $B_n$ ($n=1,2,\dots$) are given by (2.9), then $f\in M{K}^{(k)}(\beta ,\gamma )$.

Proof

Suppose that

$$G_k(z)=z^{k-1}{g}_k(z).$$

(2.8)

By Theorem 2, we know that $G_k\in M{S}^{\ast }$. Hence, equality (2.6) can be written as

$$G_k(z)=z^{k-1}{g}_k(z)=\frac{1}{z}+\sum _{n=1}^{\mathrm{\infty }}{B}_n{z}^n\in M{S}^{\ast }.$$

(2.9)

To prove $f\in M{K}^{(k)}(\beta ,\gamma )$, it suffices to show that

$$\left|\frac{\frac{z{f}^{\prime }(z)}{G_k(z)}+1}{\frac{z{f}^{\prime }(z)}{G_k(z)}+2\gamma -1}\right|<\beta ,$$

where $G_k$ is given by (2.8). From (2.7), we know that

$$\beta (2-2\gamma )-\sum _{n=1}^{\mathrm{\infty }}n\beta |a_n|-\sum _{n=1}^{\mathrm{\infty }}\beta |1-2\gamma ||B_n|\ge \sum _{n=1}^{\mathrm{\infty }}n|a_n|+\sum _{n=1}^{\mathrm{\infty }}|B_n|>0.$$

(2.10)

Now, by the maximum modulus principle, we deduce from (1.1), (2.9) and (2.10) that

$$\begin{array}{rcl}\left|\frac{\frac{z{f}^{\prime }(z)}{G_k(z)}+1}{\frac{z{f}^{\prime }(z)}{G_k(z)}+2\gamma -1}\right|& =& \left|\frac{{\sum }_{n=1}^{\mathrm{\infty }}n{a}_n{z}^{n+1}+{\sum }_{n=1}^{\mathrm{\infty }}{B}_n{z}^{n+1}}{{\sum }_{n=1}^{\mathrm{\infty }}n{a}_n{z}^{n+1}-{\sum }_{n=1}^{\mathrm{\infty }}(1-2\gamma )B_n{z}^{n+1}-(2-2\gamma )}\right|\\ <& \frac{{\sum }_{n=1}^{\mathrm{\infty }}n|a_n|+{\sum }_{n=1}^{\mathrm{\infty }}|B_n|}{(2-2\gamma )-{\sum }_{n=1}^{\mathrm{\infty }}n|a_n|-{\sum }_{n=1}^{\mathrm{\infty }}|1-2\gamma ||B_n|}\le \beta .\end{array}$$

This evidently completes the proof of Theorem 3. □

Theorem 4 Let $f(z)=\frac{1}{z}+{\sum }_{n=1}^{\mathrm{\infty }}{a}_n{z}^n\in M{K}^{(k)}(\beta ,\gamma )$. Then

$$\begin{array}{c}1+\sum _{n=1}^{\mathrm{\infty }}\frac{n(1-\beta e^{i\theta })}{2(1-\gamma )\beta e^{i\theta }}{a}_n{z}^{n+1}+\sum _{n=1}^{\mathrm{\infty }}\frac{1+(1-2\gamma )\beta e^{i\theta }}{2(1-\gamma )\beta e^{i\theta }}{B}_n{z}^{n+1}\ne 0\hfill \\ (z\in U^{\ast };0<\theta <2\pi ),\hfill \end{array}$$

(2.11)

where the coefficients $B_n$ ($n=1,2,\dots$) are given by (2.9).

Proof Suppose that $f\in M{K}^{(k)}(\beta ,\gamma )$. Then we know that

$$-\frac{z{f}^{\prime }(z)}{G_k(z)}\ne \frac{1+(1-2\gamma )\beta e^{i\theta }}{1-\beta e^{i\theta }}(z\in U^{\ast };0<\theta <2\pi ),$$

(2.12)

where $G_k$ is given by (2.8). After a simple computation, the inequality (2.2) is equivalent to

$$z{f}^{\prime }(z)(1-\beta e^{i\theta })+G_k(z)(1+(1-2\gamma )\beta e^{i\theta })\ne 0(z\in U^{\ast };0<\theta <2\pi ).$$

(2.13)

By substituting (1.1) and (2.9) into (2.13), we obtain the desired assertion (2.11) of Theorem 4.

Finally, we provide the following distortion theorem for the considered class of functions $M{K}^{(k)}(\beta ,\gamma )$. □

Theorem 5 If $f\in M{K}^{(k)}(\beta ,\gamma )$, then

$$\begin{array}{rcl}\frac{{(1-r)}^2(1-(1-2\gamma )\beta r)}{r^2(1+\beta r)}& \le & |f^{\prime }(z)|\\ \le & \frac{{(1+r)}^2(1+(1-2\gamma )\beta r)}{r^2(1-\beta r)}(|z|=r;0<r<1).\end{array}$$

(2.14)

Proof If $f\in M{K}^{(k)}(\beta ,\gamma )$, then there exists a function $g\in M{S}^{\ast }(\frac{k-1}{k})$ such that (1.3) holds true. It follows from Theorem 2 that the function $G_k$ given by (2.8) is a meromorphic starlike function. Hence, we have (see [12])

$$\frac{{(1-r)}^2}{r}\le |G_k(z)|\le \frac{{(1+r)}^2}{r}(|z|=r;0<r<1).$$

(2.15)

Let us define $p(z)$ by

$$-\frac{z{f}^{\prime }(z)}{G_k(z)}=p(z)(z\in U^{\ast }),$$

(2.16)

where

$$p(z)\prec \frac{1+(1-2\gamma )\beta z}{1-\beta z}.$$

Then, by using a similar method as in [[13], p.105], we have

$$\frac{1-(1-2\gamma )\beta r}{1+\beta r}\le |p(z)|\le \frac{1+(1-2\gamma )\beta r}{1-\beta r}(|z|=r;0<r<1).$$

(2.17)

Thus, from (2.15), (2.16) and (2.17), we readily get the inequality (2.14). The proof of Theorem 5 is thus completed. □