Let $\phi\in C^s,0<\alpha\leq s<1$, where $C^s(0,T]$ is Holder continuous functions. Is it possible to show the following inequality
$$|\frac{\phi(x)}{x^\alpha}-\frac{\phi(y)}{y^\alpha}|\leq \frac{|\phi(x)-\phi(y)|}{|x-y|^\alpha}?$$



Is it possible to show the following inequality?
The answer is no. E.g., let $\phi(u)=1$ for all $u$, and let $x\ne y$. Then the inequality fails to hold.


