Marx-Strohh\"{a}cker theorem for Multivalent Functions

TL;DR

This paper extends the classical Marx-Strohh"acker theorem to multivalent functions and generalizes results for functions with fixed second coefficients using differential subordination, correcting previous findings.

Contribution

It introduces new differential implications for multivalent functions and corrects earlier results by applying the theory of first order differential subordination.

Findings

Extended Marx-Strohh"acker theorem for multivalent functions

Generalized results for functions with fixed second coefficient

Corrected previous results by Selvaraj and Stelin

Abstract

Some differential implications of classical Marx-Strohh\"acker theorem are extended for multivalent functions. These results are also generalized for functions with fixed second coefficient by using the theory of first order differential subordination which in turn, corrects the results of Selvaraj and Stelin [On multivalent functions associated with fixed second coefficient and the principle of subordination, Int. J. Math. Anal. {\bf 9} (2015), no.~18, 883--895].