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

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