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A geometric property for a class of meromorphic analytic functions

Journal of Inequalities and Applications20142014:120

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

  • Received: 22 February 2014
  • Accepted: 13 March 2014
  • Published:

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 , z U .

Keywords

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

1 Introduction

A conformal, meromorphic function f on the punctured unit disk U ˆ : = { z C : 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 ) = 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 .
Furthermore, an analytic function f U ˆ is called a concave function of order α 0 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 , z U .

2 Main result

We have the following result.

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

then f is concave in U ˆ .

Proof To show that f is concave, we need
Re { 1 z f ( z ) f ( z ) } > 0 , z U .
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 ) = k w ( z 0 ) , where k 2 , 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 ) = k w ( z 0 ) , where k 2 , 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 + 1 3 , 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. □

Declarations

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.

Authors’ Affiliations

(1)
Institute of Mathematical Sciences, University Malaya, Kuala Lumpur, 50603, Malaysia
(2)
Department of Mathematics, Rzeszów University of Technology, Al. Powstańców Warszawy 12, Rzeszów, 35-959, Poland

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/S0002-9939-2012-11455-8MathSciNetView ArticleGoogle Scholar

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