Bounded function of several real variables

Question

Let $y>0$ be a positive real number

Let $x \in \mathbb{R}$ such that $0<x<1$

Let $a,b \in [0,x]$

I would like to know if are bounded the following functions

$$ \dfrac{1}{x} \left| \dfrac{b}{y^b} - \dfrac{a}{y^a} \right| $$ $$ \dfrac{1}{x} \left| \dfrac{1}{y^b} - \dfrac{1}{y^a} \right| $$ thanks for any suggestion.

Answer 1

For the first function take $a=0,b=x=\frac{1}{2}$ an we get

$$\sup_{a,b,x,y}\dfrac{1}{x} \left| \dfrac{b}{y^b} - \dfrac{a}{y^a} \right| \ge \sup_{y} \frac{1}{\sqrt{y}} = +\infty $$

For the second take $x=b=\frac{1}{2}, a=\frac{1}{4}$ an we get: $$\sup_{a,b,x,y} \dfrac{1}{x} \left| \dfrac{b}{y^b} - \dfrac{a}{y^a} \right| \ge \sup_{y} 2 \left|\frac{1}{2\sqrt{y}} - \frac{1}{4\sqrt[4]{y}}\right| \ge \sup_{0 < y < 1} \left|\frac{2 - \sqrt[4]{y}}{2\sqrt{y}}\right| = +\infty$$