Initial successive coefficients for certain classes of univalent functions

TL;DR

This paper investigates bounds on the differences of successive coefficients' moduli for certain classes of univalent functions, providing sharp estimates and extremal functions for the initial coefficients.

Contribution

It introduces new sharp bounds for the differences of successive coefficients' moduli in subclasses of univalent functions, with explicit extremal functions.

Findings

Sharp bounds for | |a_{n+1}| - |a_n| | for n=1,2

Construction of extremal functions achieving bounds

Results applicable to specific subclasses of univalent functions

Abstract

We consider a family of all analytic and univalent functions in the unit disk of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. The aim of this article is to investigate the bounds of the difference of moduli of initial successive coefficients, i.e. $\big | |a_{n+1}|-|a_n|\big |$ for $n=1,\,2$ and for some subclasses of analytic univalent functions. We found that all the estimations are sharp in nature by constructing some extremal functions.