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fractional power function inequality

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I am facing an issue to prove if its true the following inequality and any help is greatly appreciate. I used this inequality $x^\alpha \le \alpha x+1-\alpha$ but still I could not get the result.

Let $\alpha \in (0,1)$ and $x,y \in [0,b]$ where $b>1$ and $x<y$. Is there any constant $C$ such that $y^\alpha-x^\alpha \le C \vert x-y \vert$ ?


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