Sharp Estimates for Conjugate Functions with Applications to Trigonometric Polynomials

TL;DR

This paper establishes a sharp estimate for conjugate functions using harmonic majorants, leading to improved bounds in trigonometric polynomial inequalities and removing previous logarithmic losses.

Contribution

It introduces a novel sharp estimate for conjugate functions via harmonic majorants and applies it to optimize bounds in trigonometric polynomial theorems.

Findings

Removed the logarithmic loss in Papadopoulos's theorem

Achieved the optimal order in related inequalities

Provided a new sharp estimate for conjugate functions

Abstract

We prove a sharp estimate for conjugate functions using a harmonic majorant in a half-strip. As an application, we remove the logarithmic loss from a theorem of Papadopoulos on minima of trigonometric polynomials and obtain the optimal order in the corresponding inequality.