A simple inequality relating the Euler-Riemann zeta function, digamma, and cotangent over the unit interval

TL;DR

This paper establishes a new inequality involving the cotangent, Euler-Riemann zeta, and digamma functions, providing a simple proof and a conjecture for a stronger version, with insights into the problem's background.

Contribution

It introduces a novel inequality connecting key special functions and proposes a conjecture for its potential strengthening, along with a straightforward proof.

Findings

Proved a new inequality relating the three functions.

Provided a simple proof of the inequality.

Conjectured a possible strengthening of the inequality.

Abstract

We prove an inequality featuring three well-known functions from analysis, namely the cotangent, the Euler-Riemann zeta function, and the digamma function. Aside from a simple proof of our result, we give a conjectured strengthening. We offer various remarks about the origins of this problem.