Sharp bounds for some segments of bounded power series

TL;DR

This paper establishes precise upper bounds for specific segments of bounded power series and demonstrates that certain algebraic functions' Taylor polynomials do not have zeros inside the unit disk.

Contribution

It provides sharp bounds for three-term segments of bounded power series and proves non-vanishing of Taylor polynomials for a class of algebraic functions.

Findings

Sharp upper bounds for three-term segments of bounded power series

Taylor polynomials of certain algebraic functions do not vanish in the unit disk

Advances understanding of the behavior of bounded power series and their polynomial approximations

Abstract

We obtain sharp upper bounds for three-term segments of a bounded power series. Along the way we show that the Taylor polynomials of a certain algebraic function do not vanish in the unit disk.