A note on successive coefficients of convex functions

TL;DR

This paper studies the extremal values of differences between successive coefficients of convex functions analytic in the unit disk, providing bounds for specific coefficient functionals and exploring related maximization problems.

Contribution

It introduces new bounds for the differences of successive coefficients of convex functions and analyzes related extremal problems with fixed second derivative at zero.

Findings

Derived bounds for |a_{n+1}| - |a_{n}| for convex functions.

Solved maximization problem for |a_{n+1} - a_{n}| with fixed f''(0).

Provided insights into coefficient behavior of convex functions.

Abstract

In this note, we investigate the supremum and the infimum of the functional $|a_{n+1}|-|a_{n}|$ for functions, convex and analytic on the unit disk, of the form $f(z)=z+a_2z^2+a_3z^3+\dots.$ We also consider the related problem to maximize the functional $|a_{n+1}-a_{n}|$ for convex functions $f$ with $f"(0)=p$ for a prescribed $p\in[0,2].$