Starlikeness problems for certain analytic functions concerned with subordinations

TL;DR

This paper investigates starlikeness properties of certain analytic functions within the unit disk, focusing on functions with specific derivative conditions, and extends previous results on subordinations and starlikeness problems.

Contribution

It extends existing results on starlikeness and subordinations for functions with prescribed derivative conditions, providing new insights into their geometric properties.

Findings

Extended results on starlikeness for functions in class A_n.

Established new subordinations related to these functions.

Generalized previous theorems by Obradović.

Abstract

Let A_n be the class of functions f(z) which are analytic in the open unit disk U} with f(0)=0, f'(0)=1, f"(0)=f"'(0)=...=f^{(n)}=0 and f^{(n+1)}\neq0. Applying the results due to S. S. Miller (J. Math. Anal. Appl. 65(1978), 289-305), some interesting starlikeness problems concerned with subordinations are discussed. The results in the paper are extensions of results by M. Obradovi\'c (Hokkaido Math. J. 27(1998), 329-335).