Conjugate Exponents and Young's Inequality

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $t^{a}$, for real $t\ge0$ and $a>0$, be the real power of Real Powers.

(i) Conjugate exponents. Let $p>1$ be a real number. Then $q=p/(p-1)$ is the unique real number with

$$\frac{1}{p}+\frac{1}{q}=1,$$

and $q>1$.

(ii) Young's inequality. Let $p,q>1$ be real numbers with $1/p+1/q=1$, and let $a,b\ge0$ be real numbers. Then

$$ab\le\frac{a^{p}}{p}+\frac{b^{q}}{q} .$$