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A geometric property for a class of meromorphic analytic functions

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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 f ( z ) Open image in new window, the sufficient condition for concavity is Re { z f ( z ) f ( z ) } < 0 Open image in new window, z U Open image in new window.

Keywords

Analytic Function Geometric Property Unit Disk Meromorphic Function Conformal Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

A conformal, meromorphic function f on the punctured unit disk U ˆ : = { z C : 0 < | z | < 1 } Open image in new window is said to be a concave mapping if f ( U ˆ ) Open image in new window 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 Open image in new window of the form
f ( z ) = 1 z + b 0 + b 1 z + b 2 z 2 + , Open image in new window
(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 , Open image in new window
where
z f ( z ) f ( z ) = 2 2 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 . Open image in new window
Furthermore, an analytic function f U ˆ Open image in new window is called a concave function of order α 0 Open image in new window if it satisfies
1 + Re { z f ( z ) f ( z ) } < α , z U ˆ . Open image in new window

Denote this class by Σ α Open image in new window.

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 Open image in new window, a sufficient condition for concavity is
Re { z f ( z ) f ( z ) } < 0 , z U . Open image in new window

2 Main result

We have the following result.

Theorem 2.1 If f Σ Open image in new window satisfies the following inequality:
Re { z f ( z ) f ( z ) } < 0 , z U , Open image in new window
(2.1)
such that
Re { z f ( z ) f ( z ) } 0 , z U , Open image in new window

then f is concave in U ˆ Open image in new window.

Proof To show that f is concave, we need
Re { 1 z f ( z ) f ( z ) } > 0 , z U . Open image in new window
Let ω ( z ) Open image in new window be a function defined by
1 z f ( z ) f ( z ) = 1 + w ( z ) 1 w ( z ) . Open image in new window
(2.2)
Then w ( z ) Open image in new window is analytic in U with w ( 0 ) = w ( 0 ) = 0 Open image in new window and
z f ( z ) f ( z ) = 2 1 w ( z ) . Open image in new window
(2.3)
Therefore, we need to show that | w ( z ) | < 1 Open image in new window in U. If not, then there exists a z 0 U Open image in new window such that | w ( z 0 ) | = 1 Open image in new window. By Jack’s lemma z 0 w ( z 0 ) = k w ( z 0 ) Open image in new window, where k 2 Open image in new window, because w ( 0 ) = 0 Open image in new window. By (2.3) we have
z 3 f ( z ) ( 1 w ( z ) ) = 2 z 2 f ( z ) . Open image in new window
(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 ) , Open image in new window
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 ) Open image in new window
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 ) . Open image in new window
It gives for z = z 0 Open image in new window
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 ) . Open image in new window
By (2.3) and by z 0 w ( z 0 ) = k w ( z 0 ) Open image in new window, where k 2 Open image in new window, 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 ) . Open image in new window
Because k + 1 3 Open image in new window, a simple geometric observation yields
Re { ( k + 1 ) w ( z 0 ) 3 1 w ( z 0 ) } 0 , Open image in new window
hence
Re { z 0 f ( z 0 ) f ( z 0 ) } 0 . Open image in new window

This contradicts the assumption (2.1). Therefore, | w ( z ) | < 1 Open image in new window 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/S0002-9939-2012-11455-8MathSciNetCrossRefGoogle Scholar

Copyright information

© Ibrahim and Sokó¿; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesUniversity MalayaKuala LumpurMalaysia
  2. 2.Department of MathematicsRzeszów University of TechnologyRzeszówPoland

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