Successive coefficients of convex functions

TL;DR

This paper establishes sharp bounds on the differences of successive coefficients for convex functions analytic in the unit disk, contributing to the understanding of coefficient behavior in geometric function theory.

Contribution

It provides the first sharp bounds on the differences of successive coefficients for convex functions, specifically for |a_3| - |a_2| and |a_4| - |a_3|.

Findings

|a_3| - |a_2|  25/48

|a_4| - |a_3|  25/48

Both inequalities are proven to be sharp.

Abstract

It is shown that for $f$ analytic and convex in $z\in D=\{z:|z|<1\}$ and given by $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$, the difference of coefficients $||a_{3}|-|a_{2}||\le 25/48$ and $||a_{4}|-|a_{3}||\le 25/48$ . Both inequalities are sharp.