Proving that $x^a-y^a<x-y$ for $0<y<x<1$ and $a \in (0,1)$

Question

I have to prove this simple inequality, to complete another proof. But I don't know where to start anymore.

The inequality as in the title is: $x^a-y^a<x-y$ for $0<y<x<1$ and $a \in (0,1)$.

I already tried some stuff, through logarithms, concave functions and binomial theorem but without much success.

Thanks for your help.

Answer 1

I think that this is simply wrong: you can rewrite it as

$$x^a-x<y^a-y$$ which would mean that you want the function $x\mapsto x^a-x$ to be decreasing over $(0,1)$, except it is not.

Answer 2

It's wrong.

Try $a=\frac{1}{2}$, $x=\frac{1}{4}$ and $y=\frac{1}{5}$