The sharp estimates of all initial taylor coefficients in the Krzyz's problem

TL;DR

This paper provides exact estimates for all initial Taylor coefficients of functions within a specific class of holomorphic functions in the unit disk, extending understanding of coefficient bounds in complex analysis.

Contribution

It introduces precise estimations of initial Taylor coefficients for functions in the class $B_t$, advancing the theory of coefficient bounds in the Krzyz's problem.

Findings

Exact estimations of initial Taylor coefficients for functions in $B_t$

Determination of the number $N(t)$ up to which estimates hold

Extension of coefficient bounds in complex analysis

Abstract

For each $t>0,$ up to the number $n=N(t),$ the exact estimations of all initial taylor coefficients in the class $B_t$ were found, where $B_t$ is a set of holomorphic in unit disk functions $f,$ $0<|f|<1,$ $f(0)=e^{-t}.$