## SHARP INEQUALITIES FOR UNIVALENCE OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED UNIT DISK

## SHARP INEQUALITIES FOR UNIVALENCE OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED UNIT DISK

*O. Al-Refai*

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## Abstract

A new class of meromorphic functions f that are univalent in the punctured unit disk U∗ = {z : 0 < |z| < 1} is introduced. This class is denoted by MU and consisting of functions f defined by |1 + f′(z)/f2(z)| < 1 and zf(z) ̸= 0, whenever z ∈ U = {z : |z| < 1}. For every n ≥ 2, sharp bound for the nth derivative of 1/(zf(z)) that implies univalency of f in U∗ is established. In particular, the best improvements for known univalence criteria are obtained. Distortion and growth estimates are investigated. Further, various sufficient coefficient conditions and a necessary coefficient condition for f to be in MU are derived and best radii of univalence are obtained for certain cases.

## Keywords

univalent functions, meromorphic functions, distortion theorem, coefficient bounds, area theorem./p>