I want to find the the supremum of $\dfrac{|x|^{2/3}-|y|^{2/3}}{|x-y|^{2/3}} $ in the unit ball centered at the origin . Here $x\neq y$, $x,y \in \mathbb R^n$. How do I proceed ? Thank you for your help.
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Hint: By a modification of the triangular inequality you see that $$||x|^{2/3}-|y|^{2/3}|\leq |x-y|^{2/3}$$ for all $x\neq y$. So we have$$\dfrac{|x|^{2/3}-|y|^{2/3}}{|x-y|^{2/3}}\leq 1$$ Is the value of the function $=1$ at some point? |
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