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# A geometric property for a class of meromorphic analytic functions

*Journal of Inequalities and Applications*
**volume 2014**, Article number: 120 (2014)

## Abstract

In this paper, we investigate a geometric property of a class of meromorphic functions. This property implies concavity. A sufficient condition, for a function in this class, is considered utilizing Jack’s lemma. We show that, for a meromorphic function f(z), the sufficient condition for concavity is \mathfrak{Re}\{\frac{z{f}^{\u2034}(z)}{{f}^{\u2033}(z)}\}<0, z\in U.

## 1 Introduction

A conformal, meromorphic function *f* on the punctured unit disk \stackrel{\u02c6}{U}:=\{z\in \mathbb{C}:0<|z|<1\} is said to be a concave mapping if f(\stackrel{\u02c6}{U}) is the complement of a convex, compact set. Recently, Chuaqui *et al.* [1] studied the normalized conformal mappings of the disk onto the exterior of a convex polygon via an exemplification formula furnished by the Schwarz lemma. Let Σ be the family of functions analytic in the punctured unit disk \stackrel{\u02c6}{U} of the form

then the necessary and sufficient condition for *f* to be concave mapping is

where

Furthermore, an analytic function f\in \stackrel{\u02c6}{U} is called a concave function of order \alpha \ge 0 if it satisfies

Denote this class by {\mathrm{\Sigma}}_{\alpha}.

In this work, we investigate a geometric property of a class of meromorphic functions. This property implies concavity. A sufficient condition, for a function in this class, is considered utilizing Jack’s lemma. We show that, for a meromorphic function f(z)\in \mathrm{\Sigma}, a sufficient condition for concavity is

## 2 Main result

We have the following result.

**Theorem 2.1** *If* f\in \mathrm{\Sigma} *satisfies the following inequality*:

*such that*

*then* *f* *is concave in* \stackrel{\u02c6}{U}.

*Proof* To show that *f* is concave, we need

Let \omega (z) be a function defined by

Then w(z) is analytic in *U* with w(0)={w}^{\prime}(0)=0 and

Therefore, we need to show that |w(z)|<1 in *U*. If not, then there exists a {z}_{0}\in U such that |w({z}_{0})|=1. By Jack’s lemma {z}_{0}{w}^{\prime}({z}_{0})=kw({z}_{0}), where k\ge 2, because {w}^{\prime}(0)=0. By (2.3) we have

Differentiating logarithmically (2.4) with respect to *z*, we conclude

hence

and

It gives for z={z}_{0}

By (2.3) and by {z}_{0}{w}^{\prime}({z}_{0})=kw({z}_{0}), where k\ge 2, we have

Because k+1\ge 3, a simple geometric observation yields

hence

This contradicts the assumption (2.1). Therefore, |w(z)|<1 in *U* and (2.2) means that *f* is concave. □

## References

Chuaqui M, Duren P, Osgood B:

**Concave conformal mappings and pre-vertices of Schwarz-Christoffel mappings.***Proc. Am. Math. Soc.*2012,**140:**3495–3505. 10.1090/S0002-9939-2012-11455-8

## Acknowledgements

This work is supported by University of Malaya High Impact Research Grant no vote UM.C/625/HIR/MOHE/SC/13/2 from Ministry of Higher Education Malaysia. The authors also would like to thank the referees for giving useful suggestions for improving the work.

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### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Both authors jointly worked on deriving the results and approved the final manuscript.

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### Cite this article

Ibrahim, R.W., Sokół, J. A geometric property for a class of meromorphic analytic functions.
*J Inequal Appl* **2014**, 120 (2014). https://doi.org/10.1186/1029-242X-2014-120

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DOI: https://doi.org/10.1186/1029-242X-2014-120

### Keywords

- Analytic Function
- Geometric Property
- Unit Disk
- Meromorphic Function
- Conformal Mapping