Separation of variables and inequality

Question

I need to separate variables x, w and y,z to obtain in the right hand of inequality the product of two linear form with x and w in one hand and y and z in the other hand: Assume $\alpha \leq 0$, $x, w\in \mathbb{R} $ and $y, z \in (0,1) $ $\vert x\sqrt{y}-w\sqrt{z}\vert ^{\alpha} \leq F(x,w)G(y,z)$

Any ideas of type of inequality i can use ?

Answer 1

CS inequality would give you $$|x\sqrt y - w \sqrt z|^2 \leqslant (x^2+w^2)(y + z)$$ $$\implies |x\sqrt y - w \sqrt z|^\alpha \leqslant (x^2+w^2)^{\alpha/2}(y + z)^{\alpha/2}$$