Inequalities for finite trigonometric sums. An interplay: with some series related to Harmonic numbers

TL;DR

This paper explores the relationship between series involving Harmonic numbers and finite trigonometric sums, deriving sharp inequalities and solving an open problem in the field.

Contribution

It establishes a novel connection between Harmonic number series and trigonometric sums, providing new bounds and resolving an open question by H. Chen.

Findings

Derived sharp inequalities for finite trigonometric sums.

Expressed series involving Harmonic numbers in terms of trigonometric sums.

Solved an open problem posed by H. Chen in 2010.

Abstract

An interplay between the sum of certain series related to Harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits us to find sharp inequalities bounding these trigonometric sums. In particular, this answers positively an open problem of H. Chen (2010).