Close-to-convexity and starlikeness of analytic functions

TL;DR

This paper investigates conditions under which certain analytic functions are close-to-convex, focusing on the role of the real part of their derivatives, and explores connections with existing results in geometric function theory.

Contribution

It introduces new sufficient conditions for close-to-convexity of analytic functions based on the real part of their derivatives, extending previous work in the field.

Findings

Established a sufficient condition for close-to-convexity involving the derivative's real part.

Connected the new conditions with previously known results in geometric function theory.

Provided several examples illustrating the applicability of the conditions.

Abstract

For functions $f(z)=z^p+a_{n+1}z^{p+1}+...$ defined on the open unit disk, the condition $\Re (f'(z)/z^{p-1})>0$ is sufficient for close-to-convexity of $f$. By making use of this result, several sufficient conditions for close-to-convexity are investigated and relevant connections with previously known results are indicated.