A Criterion for Univalent Meromorphic Functions

Abstract

\begin {abstract}

Let $\mathbb{D} = \{ z \in \mathbb{C}, | z | < 1 \} $ and $\mathcal{A} ( p )$ be the set of meromorphic functions in

$\mathbb{D}$ possessing one and simple pole at the point $p$ with $p \in \, ( 0 \, , \, 1 )$.\\

The aim of this paper is to give a criterion by mean of conditions on the parameters $\a , \b \, \in \mathbb{C}$,

$\l > 0$ and

$g \, \in \, \mathcal{A} (p ) $ for functions in the class denoted $\mathcal{P}_{ \a, \b\, ; h } ( p \, ; \, \l )$ of functions $f$ $\mathcal{A} (p ) $ satisfying a differential Inequality of the form :

$$| \a ( \frac{z}{f ( z )} )'' + \b \, ( \frac{z}{g ( z )} )'' | \, \leq \l \m, \; z \in \mathbb{D} $$

to be univalent in the disc $\mathbb{D}$, where $\m = ( \frac{1 - p}{1 + p})^2$ .

\end{abstract}