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 $p, q, \alpha \in (0, \infty)$ 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} \sum_{n = 1}^N \lambda_n x_n y_n \leq \frac{\alpha^p}{p} \sum_{n = 1}^N \lambda_n x^p_n + \frac{1}{q \alpha^q} \sum_{n = 1}^N \lambda_n y^q_n \end{equation}
(2)	\begin{equation} \sum_{n = 1}^N \lambda_n x_n y_n = \frac{\alpha^p}{p} \sum_{n = 1}^N \lambda_n x^p_n + \frac{1}{q \alpha^q} \sum_{n = 1}^N \lambda_n y^q_n \quad \iff \quad \forall \, 1 \leq n \leq N : \frac{\alpha^p}{p} x^p_n = \frac{1}{q \alpha^q} y^q_n \end{equation}