Let $A,B,\alpha>0$ and $0\leq x<y$. I am currently stuck at trying to determine whether there is a constant $D > 0$ such that $$Ay^\alpha - Bx^\alpha\leq D(y^\alpha - x^\alpha)$$ for all $0\leq x < y$. As $Ay^\alpha - Bx^\alpha\geq 0\Longleftrightarrow \left(\frac{A}{B}\right)^{1/\alpha}\geq \frac{x}{y}$, we can assume that $A\geq B$. I was trying naive to estimate
$$\frac{Ay^\alpha - Bx^\alpha}{y^\alpha - x^\alpha}\leq D$$
whence one gets
$$A + B + A\frac{x^\alpha}{y^\alpha - x^\alpha} - B\frac{y^\alpha}{y^\alpha - x^\alpha}\leq D$$
When $Ax^\alpha - By^\alpha \leq 0 \Longleftrightarrow x\leq \left(\frac{B}{A}\right)^{1/\alpha}y$, the constant $D$ can be taken to be just $A + B$. But I don't know how to conclude the case when $\left(\frac{B}{A}\right)^{1/\alpha}y < x < y$.