Skip to main content

A geometric property for a class of meromorphic analytic functions

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 Re{ z f ( z ) f ( z ) }<0, zU.

1 Introduction

A conformal, meromorphic function f on the punctured unit disk U ˆ :={zC:0<|z|<1} is said to be a concave mapping if f( U ˆ ) 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 U ˆ of the form

f(z)= 1 z + b 0 + b 1 z+ b 2 z 2 +,
(1.1)

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

1+Re { z f ( z ) f ( z ) } <0,z U ˆ ,

where

z f ( z ) f ( z ) =22 b 1 z 2 6 b 2 z 3 ( 12 b 3 + 2 b 1 2 ) z 4 (20 b 4 +10 b 1 b 2 ) z 5 .

Furthermore, an analytic function f U ˆ is called a concave function of order α0 if it satisfies

1+Re { z f ( z ) f ( z ) } <α,z U ˆ .

Denote this class by Σ α .

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)Σ, a sufficient condition for concavity is

Re { z f ( z ) f ( z ) } <0,zU.

2 Main result

We have the following result.

Theorem 2.1 If fΣ satisfies the following inequality:

Re { z f ( z ) f ( z ) } <0,zU,
(2.1)

such that

Re { z f ( z ) f ( z ) } 0,zU,

then f is concave in U ˆ .

Proof To show that f is concave, we need

Re { 1 z f ( z ) f ( z ) } >0,zU.

Let ω(z) be a function defined by

1 z f ( z ) f ( z ) = 1 + w ( z ) 1 w ( z ) .
(2.2)

Then w(z) is analytic in U with w(0)= w (0)=0 and

z f ( z ) f ( z ) = 2 1 w ( z ) .
(2.3)

Therefore, we need to show that |w(z)|<1 in U. If not, then there exists a z 0 U such that |w( z 0 )|=1. By Jack’s lemma z 0 w ( z 0 )=kw( z 0 ), where k2, because w (0)=0. By (2.3) we have

z 3 f (z) ( 1 w ( z ) ) =2 z 2 f (z).
(2.4)

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

3 z 2 f ( z ) + z 3 f ( z ) z 3 f ( z ) w ( z ) 1 w ( z ) = 2 z f ( z ) + z 2 f ( z ) z 2 f ( z ) ,

hence

3 z 3 f ( z ) + z 4 f ( z ) z 3 f ( z ) z w ( z ) 1 w ( z ) = 2 z 2 f ( z ) + z 3 f ( z ) z 2 f ( z )

and

3+ z 4 f ( z ) z 3 f ( z ) z w ( z ) 1 w ( z ) =2+ z 3 f ( z ) z 2 f ( z ) .

It gives for z= z 0

3+ z 0 f ( z 0 ) f ( z 0 ) z 0 w ( z 0 ) 1 w ( z 0 ) =2+ z 0 f ( z 0 ) f ( z 0 ) .

By (2.3) and by z 0 w ( z 0 )=kw( z 0 ), where k2, we have

z 0 f ( z 0 ) f ( z 0 ) = z 0 w ( z 0 ) 1 w ( z 0 ) 1 + z 0 f ( z 0 ) f ( z 0 ) = z 0 w ( z 0 ) 1 w ( z 0 ) 1 2 1 w ( z ) = k w ( z 0 ) 1 w ( z 0 ) 1 2 1 w ( z ) = ( k + 1 ) w ( z 0 ) 3 1 w ( z 0 ) .

Because k+13, a simple geometric observation yields

Re { ( k + 1 ) w ( z 0 ) 3 1 w ( z 0 ) } 0,

hence

Re { z 0 f ( z 0 ) f ( z 0 ) } 0.

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

References

  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

    MathSciNet  Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rabha W Ibrahim.

Additional information

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.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

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