On a Sine Polynomial of Turan

TL;DR

This paper explores inequalities related to a sine polynomial originally studied by Turán, providing refinements and applications to Chebyshev polynomials of the second kind.

Contribution

The paper introduces new inequalities for Turán's sine polynomial and applies these results to derive inequalities for Chebyshev polynomials.

Findings

Refined inequalities for sine sums involving Turán's polynomial

Validation of inequalities for specific parameter ranges

Application of sine sum inequalities to Chebyshev polynomials

Abstract

In 1935, P. Tur\'an proved that $$ S_{n,a}(x)= \sum_{j=1}^n{n+a-j\choose n-j} \sin(jx)>0 \quad{(n,a\in\mathbf{N}; 0<x<\pi).} $$ We present various related inequalities. Among others, we show that the refinements $$ S_{2n-1,a}(x)\geq \sin(x) \quad\mbox{and} \quad{S_{2n,a}(x)\geq 2\sin(x)(1+\cos(x))} $$ are valid for all integers $n\geq 1$ and real numbers $a\geq 1$ and $x\in(0,\pi)$. Moreover, we apply our theorems on sine sums to obtain inequalities for the Chebyshev polynomials of the second kind.