Extensions of Nunokawa lemma for argument properties

TL;DR

This paper extends Nunokawa's lemma for functions analytic in the unit disk, providing new results on argument properties and broadening the scope of previous theorems.

Contribution

It introduces generalized extensions of Nunokawa's lemma and applies them to analyze argument properties of analytic functions.

Findings

Extended Nunokawa lemma for broader classes of functions

New theorems on argument properties derived from the extensions

Applications demonstrating the utility of the extended lemma

Abstract

Let H[a_0,n] be the class of functions p(z)=a_0+a_nz^n+... which are analytic in the open unit disk U. For p(z)in H[1,2], M. Nunokawa, S. Owa, N. Uyanik and H. Shiraishi (Math. Comput. Modelling. 55 (2012), 1245-1250) have shown some theorems for argument properties. The object of the present paper is to discuss some extensions of Nnnokawa lemma and its applications for argument properties.