# A geometric property for a class of meromorphic analytic functions

Open Access
Research

## 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 , the sufficient condition for concavity is , .

### Keywords

Analytic Function Geometric Property Unit Disk Meromorphic Function Conformal Mapping

## 1 Introduction

A conformal, meromorphic function f on the punctured unit disk is said to be a concave mapping if 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 of the form
(1.1)
then the necessary and sufficient condition for f to be concave mapping is
where
Furthermore, an analytic function is called a concave function of order if it satisfies

Denote this class by .

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 , a sufficient condition for concavity is

## 2 Main result

We have the following result.

Theorem 2.1 If satisfies the following inequality:
(2.1)
such that

then f is concave in .

Proof To show that f is concave, we need
Let be a function defined by
(2.2)
Then is analytic in U with and
(2.3)
Therefore, we need to show that in U. If not, then there exists a such that . By Jack’s lemma , where , because . By (2.3) we have
(2.4)
Differentiating logarithmically (2.4) with respect to z, we conclude
hence
and
It gives for
By (2.3) and by , where , we have
Because , a simple geometric observation yields
hence

This contradicts the assumption (2.1). Therefore, in U and (2.2) means that f is concave. □

## Notes

### 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.

### References

1. 1.
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