Let $x, y \in [0, \infty)$ each be an D4767: Unsigned real number.
Let $p, q, \alpha \in (0, \infty)$ each be a D5407: Positive real number such that

(i)	\begin{equation} \frac{1}{p} + \frac{1}{q} = 1 \end{equation}

Then

(1)	\begin{equation} x y \leq \frac{\alpha^p}{p} x^p + \frac{1}{q \alpha^q} y^q \end{equation}
(2)	\begin{equation} x y = \frac{\alpha^p}{p} x^p + \frac{1}{q \alpha^q} y^q \quad \iff \quad \frac{\alpha^p}{p} x^p = \frac{1}{q \alpha^q} y^q \end{equation}