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 $$|\frac{\phi(x)}{x^\alpha}-\frac{\phi(y)}{y^\alpha}|\leq \frac{|\phi(x)-\phi(y)|}{|x-y|^\alpha}?$$
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The answer is no. E.g., let $\phi(u)=1$ for all $u$, and let $x\ne y$. Then the inequality fails to hold.
answered May 20, 2021 at 14:26
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3$\begingroup$ More generally, a good first step in assessing the viability of any inequality involving non-negative quantities is to check that the vanishing of the RHS implies the vanishing of the LHS. (This is obviously a necessary condition for the inequality to hold, though it is far from sufficient.) $\endgroup$ Commented May 20, 2021 at 17:20
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$\begingroup$ Thanks for you comment. This is much useful. $\endgroup$ Commented May 22, 2021 at 11:48