SHARP INEQUALITIES FOR UNIVALENCE OF MEROMORPHIC FUNCTIONS IN THE PUNCTURED UNIT DISK - Al-Refai, Oqlah (TWMS journal of applied and engineering mathematics, 2021)
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)/f (2)(z)| < 1 and zf (z) [not equal to] 0, whenever z [member of] U = z : |z| < 1. For every n [greater than or equal to] 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. AMS Subject Classification: 30C45, 30C55