Conjugate Exponents and Young's Inequality

In the setting of The Real Numbers: Standing Notation and Background, write $\mathbb{R}_{+}$ for the set of nonnegative real numbers and, for $t\in\mathbb{R}_{+}$ and a positive real $a$, write $t^{a}$ for the power of $t$ with exponent $a$, whose properties are those of Properties of Real Powers of Nonnegative Real Numbers. Then the following hold.

1. (Conjugate exponents)¶ Let $p$ be a real number with $1<p$. There is exactly one real number $q$ with

namely $q=\dfrac{p}{p-1}$. It satisfies $1<q$, $(p-1)q=p$ and $p+q=pq$; and $p$ is in turn the unique real number bearing this relation to $q$. Two real numbers $p,q$ related in this way are called conjugate exponents.

2. (Young's inequality)¶ Let $p,q$ be conjugate exponents as in claim 1 and let $a,b\in\mathbb{R}_{+}$. Then