Certain Estimates of Normalized Analytic Functions

TL;DR

This paper investigates distortion and growth theorems, as well as bounds on coefficients and Hankel determinants, for a class of normalized analytic functions subordinate to a convex function, extending classical results in geometric function theory.

Contribution

It introduces a unified class of functions satisfying a second order differential subordination and derives new bounds and theorems for this class.

Findings

Distortion and growth theorems established for the class.

Bounds on initial logarithmic coefficients and inverse coefficients derived.

Estimates for the second Hankel determinant involving inverse coefficients obtained.

Abstract

Let $\phi$ be a normalized convex function defined on open unit disk $\mathbb{D}$. For a unified class of normalized analytic functions which satisfy the second order differential subordination $f'(z)+ \alpha z f''(z) \prec \phi(z)$ for all $z\in \mathbb{D}$, we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.