On the simultaneous Approximation of coefficients of schlicht functions

TL;DR

This paper investigates the conditions under which the coefficients of schlicht functions, normalized by n, uniformly approach their Hayman Indexes, using a modified Hardy-Littlewood Tauberian Theorem.

Contribution

It introduces a modified Hardy-Littlewood Tauberian Theorem to analyze the asymptotic behavior of schlicht function coefficients.

Findings

Coefficients |a(n)/n| tend uniformly to Hayman Indexes under certain conditions.

The modified Tauberian Theorem provides a new tool for coefficient approximation.

Conditions for uniform convergence are explicitly characterized.

Abstract

A modified Version of the Hardy-Littlewood tauberian Theorem is used to prove under which conditions the moduli of the coefficients |a(n)/n| of schlicht functions tend uniformly to their Hayman Indexes as n tends to infinity.