Starlikeness Using Special Functions and Subordination

TL;DR

This paper investigates conditions under which certain analytic functions are subordinate to functions with positive real part, using special functions and subordination, and establishes starlikeness of well-known functions through differential subordination techniques.

Contribution

It introduces new sharp estimates for parameter $eta$ ensuring subordination and employs admissibility to handle higher order differential subordination, advancing the theory of starlikeness.

Findings

Derived sharp bounds on $eta$ for subordination.

Established second and third order differential subordination relations.

Proved starlikeness of several classical functions.

Abstract

The association of subordination and special functions is used to find sharp estimates on the parameter $\beta$ such that the analytic function $p(z)$ is subordinate to certain functions having positive real part whenever $p(z)+\beta z p'(z)$ is subordinate to the Janowski function. Further, when the traditional approach of solving higher order differential subordination implications failed, the concept of admissibility is employed to establish certain second and third order differential subordination relations between the analytic function $p$ and the functions associated with right half plane. As a sequel, we demonstrated the starlikeness of various well-known analytic functions as well.