# 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 $f\left(z\right)$, the sufficient condition for concavity is $\mathfrak{Re}\left\{\frac{z{f}^{‴}\left(z\right)}{{f}^{″}\left(z\right)}\right\}<0$, $z\in U$.

## 1 Introduction

A conformal, meromorphic function f on the punctured unit disk $\stackrel{ˆ}{U}:=\left\{z\in \mathbb{C}:0<|z|<1\right\}$ is said to be a concave mapping if $f\left(\stackrel{ˆ}{U}\right)$ 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 $\stackrel{ˆ}{U}$ of the form

$f\left(z\right)=\frac{1}{z}+{b}_{0}+{b}_{1}z+{b}_{2}{z}^{2}+\cdots ,$
(1.1)

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

$1+\mathfrak{Re}\left\{z\frac{{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right\}<0,\phantom{\rule{1em}{0ex}}z\in \stackrel{ˆ}{U},$

where

$z\frac{{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}=-2-2{b}_{1}{z}^{2}-6{b}_{2}{z}^{3}-\left(12{b}_{3}+2{b}_{1}^{2}\right){z}^{4}-\left(20{b}_{4}+10{b}_{1}{b}_{2}\right){z}^{5}-\cdots .$

Furthermore, an analytic function $f\in \stackrel{ˆ}{U}$ is called a concave function of order $\alpha \ge 0$ if it satisfies

$1+\mathfrak{Re}\left\{z\frac{{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right\}<-\alpha ,\phantom{\rule{1em}{0ex}}z\in \stackrel{ˆ}{U}.$

Denote this class by ${\mathrm{\Sigma }}_{\alpha }$.

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 $f\left(z\right)\in \mathrm{\Sigma }$, a sufficient condition for concavity is

$\mathfrak{Re}\left\{\frac{z{f}^{‴}\left(z\right)}{{f}^{″}\left(z\right)}\right\}<0,\phantom{\rule{1em}{0ex}}z\in U.$

## 2 Main result

We have the following result.

Theorem 2.1 If $f\in \mathrm{\Sigma }$ satisfies the following inequality:

$\mathfrak{Re}\left\{\frac{z{f}^{‴}\left(z\right)}{{f}^{″}\left(z\right)}\right\}<0,\phantom{\rule{1em}{0ex}}z\in U,$
(2.1)

such that

$\mathfrak{Re}\left\{\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right\}\ne 0,\phantom{\rule{1em}{0ex}}z\in U,$

then f is concave in $\stackrel{ˆ}{U}$.

Proof To show that f is concave, we need

$\mathfrak{Re}\left\{-1-\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}\right\}>0,\phantom{\rule{1em}{0ex}}z\in U.$

Let $\omega \left(z\right)$ be a function defined by

$-1-\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}=\frac{1+w\left(z\right)}{1-w\left(z\right)}.$
(2.2)

Then $w\left(z\right)$ is analytic in U with $w\left(0\right)={w}^{\prime }\left(0\right)=0$ and

$\frac{z{f}^{″}\left(z\right)}{{f}^{\prime }\left(z\right)}=\frac{-2}{1-w\left(z\right)}.$
(2.3)

Therefore, we need to show that $|w\left(z\right)|<1$ in U. If not, then there exists a ${z}_{0}\in U$ such that $|w\left({z}_{0}\right)|=1$. By Jack’s lemma ${z}_{0}{w}^{\prime }\left({z}_{0}\right)=kw\left({z}_{0}\right)$, where $k\ge 2$, because ${w}^{\prime }\left(0\right)=0$. By (2.3) we have

$-{z}^{3}{f}^{″}\left(z\right)\left(1-w\left(z\right)\right)=2{z}^{2}{f}^{\prime }\left(z\right).$
(2.4)

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

$\frac{3{z}^{2}{f}^{″}\left(z\right)+{z}^{3}{f}^{‴}\left(z\right)}{{z}^{3}{f}^{″}\left(z\right)}-\frac{{w}^{\prime }\left(z\right)}{1-w\left(z\right)}=\frac{2z{f}^{\prime }\left(z\right)+{z}^{2}{f}^{″}\left(z\right)}{{z}^{2}{f}^{\prime }\left(z\right)},$

hence

$\frac{3{z}^{3}{f}^{″}\left(z\right)+{z}^{4}{f}^{‴}\left(z\right)}{{z}^{3}{f}^{″}\left(z\right)}-\frac{z{w}^{\prime }\left(z\right)}{1-w\left(z\right)}=\frac{2{z}^{2}{f}^{\prime }\left(z\right)+{z}^{3}{f}^{″}\left(z\right)}{{z}^{2}{f}^{\prime }\left(z\right)}$

and

$3+\frac{{z}^{4}{f}^{‴}\left(z\right)}{{z}^{3}{f}^{″}\left(z\right)}-\frac{z{w}^{\prime }\left(z\right)}{1-w\left(z\right)}=2+\frac{{z}^{3}{f}^{″}\left(z\right)}{{z}^{2}{f}^{\prime }\left(z\right)}.$

It gives for $z={z}_{0}$

$3+\frac{{z}_{0}{f}^{‴}\left({z}_{0}\right)}{{f}^{″}\left({z}_{0}\right)}-\frac{{z}_{0}{w}^{\prime }\left({z}_{0}\right)}{1-w\left({z}_{0}\right)}=2+\frac{{z}_{0}{f}^{″}\left({z}_{0}\right)}{{f}^{\prime }\left({z}_{0}\right)}.$

By (2.3) and by ${z}_{0}{w}^{\prime }\left({z}_{0}\right)=kw\left({z}_{0}\right)$, where $k\ge 2$, we have

$\begin{array}{rcl}\frac{{z}_{0}{f}^{‴}\left({z}_{0}\right)}{{f}^{″}\left({z}_{0}\right)}& =& \frac{{z}_{0}{w}^{\prime }\left({z}_{0}\right)}{1-w\left({z}_{0}\right)}-1+\frac{{z}_{0}{f}^{″}\left({z}_{0}\right)}{{f}^{\prime }\left({z}_{0}\right)}\\ =& \frac{{z}_{0}{w}^{\prime }\left({z}_{0}\right)}{1-w\left({z}_{0}\right)}-1-\frac{2}{1-w\left(z\right)}\\ =& \frac{kw\left({z}_{0}\right)}{1-w\left({z}_{0}\right)}-1-\frac{2}{1-w\left(z\right)}\\ =& \frac{\left(k+1\right)w\left({z}_{0}\right)-3}{1-w\left({z}_{0}\right)}.\end{array}$

Because $k+1\ge 3$, a simple geometric observation yields

$\mathfrak{Re}\left\{\frac{\left(k+1\right)w\left({z}_{0}\right)-3}{1-w\left({z}_{0}\right)}\right\}\ge 0,$

hence

$\mathfrak{Re}\left\{\frac{{z}_{0}{f}^{‴}\left({z}_{0}\right)}{{f}^{″}\left({z}_{0}\right)}\right\}\ge 0.$

This contradicts the assumption (2.1). Therefore, $|w\left(z\right)|<1$ 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

## 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

### Corresponding author

Correspondence to Rabha W Ibrahim.

### 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.

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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