Improved Vietoris Sine Inequalities for Non-Monotone, Non-Decaying Coefficients

TL;DR

This paper advances Vietoris sine inequalities by allowing non-monotone, non-decaying coefficients in sine polynomials, providing new extremal sum results that broaden the inequality's applicability.

Contribution

The paper introduces two improved inequalities for sine polynomials with non-monotone, non-decaying coefficients, extending previous results and identifying extremal coefficient sequences.

Findings

Derived new extremal coefficient sequences for sine inequalities.

Extended Vietoris sine inequalities to non-monotone, non-decaying coefficients.

Provided explicit coefficient sequences demonstrating the improvements.

Abstract

Recently the author established an improvement of the classical Vietoris sine inequality to include sine polynomials with non-monotone coefficients. In this paper two further improvements are presented admitting sine polynomials with non-monotone and non-decaying coefficients. The extremal sums of the two results have the coefficient sequences {2a, a, 4/3, 1, 6/5, 1, 8/7, 1, ...}, where a = 0.78265..., and {3, 3/2, 7/3, 7/4, 11/5, 11/6, ...}.