Let $x_1, y_1, \ldots, x_N, y_N \in [0, \infty)$ each be an D4767: Unsigned real number.
Let $\lambda_1, \ldots, \lambda_N \in [0, \infty)$ each be an D4767: Unsigned real number such that

(i)	\begin{equation} \sum_{n = 1}^N \lambda_n = 1 \end{equation}

Let $p, q \in (0, \infty)$ 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} \sum_{n = 1}^N \lambda_n x_n y_n \leq \left( \sum_{n = 1}^N \lambda_n x_n^p \right)^{1 / p} \left( \sum_{n = 1}^N \lambda_n y_n^q \right)^{1 / q} \end{equation}
(2)	\begin{equation} \sum_{n = 1}^N \lambda_n x_n y_n = \left( \sum_{n = 1}^N \lambda_n x_n^p \right)^{1 / p} \left( \sum_{n = 1}^N \lambda_n y_n^q \right)^{1 / q} \quad \iff \quad \exists \, c \in [0, \infty) : x = c y \text{ or } y = c x \end{equation}