An application of Miller and Mocanu lemma

TL;DR

This paper explores the application of Miller and Mocanu lemma to functions in the class H[a_0,n], generalizing Jack's lemma, and discusses properties and examples within this context.

Contribution

It applies Miller and Mocanu lemma to analyze functions in H[a_0,n], extending the understanding of their properties and providing illustrative examples.

Findings

Derived new properties for functions in H[a_0,n]

Extended Jack's lemma through Miller and Mocanu lemma

Provided examples illustrating the theoretical results

Abstract

Let H[a_0,n] be the class of functions f(z)=a_0+a_nz^n+...which are analytic in the open unit disk U}. For f(z) in H[a_0,n], S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65(1978), 289-305) have shown Miller and Mocanu lemma which is the generalization of Jack lemma by I. S. Jack (J. London Math. Soc. 3(1971), 469-474). Applying Miller and Mocanu lemma, an interesting property for f(z) in H[a_0,n] and an example are discuss.