The function $f$ with domain $x>0$ is defined by $f(x)=\frac{1}{a} x^{a}$, where $a>1$. Verify that $f$ is convex, using an appropriate sufficient condition.

Determine the Legendre transform $f^{*}$ of $f$, specifying clearly its domain of definition, and find $\left(f^{*}\right)^{*}$.

Show that

$$\frac{x^{r}}{r}+\frac{y^{s}}{s} \geqslant x y$$

where $x, y>0$ and $r$ and $s$ are positive real numbers such that $\frac{1}{r}+\frac{1}{s}=1$.