A geometric property for a class of meromorphic analytic functions
Journal of Inequalities and Applications volume 2014, Article number: 120 (2014)
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 , .
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.  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
then the necessary and sufficient condition for f to be concave mapping is
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:
then f is concave in .
Proof To show that f is concave, we need
Let be a function defined by
Then is analytic in U with and
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
Differentiating logarithmically (2.4) with respect to z, we conclude
It gives for
By (2.3) and by , where , we have
Because , a simple geometric observation yields
This contradicts the assumption (2.1). Therefore, in U and (2.2) means that f is concave. □
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
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.
The authors declare that they have no competing interests.
Both authors jointly worked on deriving the results and approved the final manuscript.
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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