A generalization of Livingston's coefficient inequalities for functions with positive real part

TL;DR

This paper extends Livingston's coefficient inequalities for functions with positive real part, providing sharper bounds, simplified proofs, and applications to holomorphic self-maps of the unit disk.

Contribution

It generalizes and simplifies Livingston's inequalities, introduces sharp bounds involving coefficients and determinants, and applies these results to holomorphic functions.

Findings

Established new sharp bounds for coefficients of functions with positive real part.

Provided simplified proofs of existing inequalities and theorems.

Applied results to inequalities for holomorphic self-maps of the unit disk.

Abstract

For functions $p(z) = 1 + \sum_{n=1}^\infty p_n z^n$ holomorphic in the unit disk, satisfying $ {\rm Re}\, p(z) > 0$, we generalize two inequalities proved by Livingston in 1969 and 1985, and simplify their proofs. One of our results states that $|p_n -w p_k p_{n-k}|\leq 2\max\{1, |1-2w|\}, w\in\mathbb{C}$. Another result involves certain determinants whose entries are the coefficients $p_n$. Both results are sharp. As applications we provide a simple proof of a theorem of J.E. Brown and various inequalities for the coefficients of holomorphic self-maps of the unit disk.