Note on the H\"{o}lder norm estimate of the function $x\sin(1/x)$
  In this paper, we prove the following inequality: for any $x, y>0$, there holds $$\big|x\sin\frac{1}{x} - y\sin\frac{1}{y} \big| \leq \sqrt{2|x - y|}.$$ 
Jiaqiang Mei, Haifeng Xu
math.CA