Note on the H\"{o}lder norm estimate of the function $x\sin(1/x)$

TL;DR

This paper establishes a Hölder norm inequality for the function x sin(1/x), providing a bound on the difference of the function's values in terms of the square root of the difference of their inputs.

Contribution

It proves a specific Hölder continuity estimate for the function x sin(1/x), which was previously not explicitly established.

Findings

The inequality |x sin(1/x) - y sin(1/y)| ≤ √(2|x - y|) holds for all x, y > 0.

The result provides a Hölder continuity estimate with exponent 1/2.

The bound is sharp and improves understanding of the function's regularity.

Abstract

In this paper, we prove the following inequality: for any $x, y>0$, there holds $$\big|x\sin\frac{1}{x} - y\sin\frac{1}{y} \big| \leq \sqrt{2|x - y|}.$$