Coefficient estimates for Schwarz functions

TL;DR

This paper derives new bounds on the coefficients of Schwarz functions, which are analytic functions bounded by 1 in the unit disk, using a lemma by Livingston.

Contribution

It introduces novel coefficient estimates for Schwarz functions leveraging Livingston's lemma, advancing understanding of their analytic properties.

Findings

New coefficient bounds for Schwarz functions.

Application of Livingston's lemma to estimate coefficients.

Enhanced understanding of Schwarz functions' analytic behavior.

Abstract

Let $\mathcal{B}$ be the class of functions $w(z)$ of the form $w(z)=\sum\limits_{k=1}^{\infty}b_k z^k$ which are analytic and satisfy the condition $|w(z)|<1$ in the open unit disk $\mathbb{U}=\left\{z\in \mathbb{C}:|z|<1\right\}$. Then we call $w(z)\in \mathcal{B}$ the Schwarz function. In this paper, we discuss new coefficient estimates for Schwarz functions by applying the lemma due to Livingston (Proc. Amer. Math. Soc. 21(1969), 545--552).