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}
|
Let $n = 1, \ldots, N$. Using result
R5188: and multiplying both sides by $\lambda_n$, we have the inequality
\begin{equation}
\lambda_n x_n y_n
\leq \frac{\alpha^p}{p} \lambda_n x^p_n + \frac{1}{q \alpha^q} \lambda_n y^q_n
\end{equation}
with equality if and only if $\frac{\alpha^p}{p} x^p_n = \frac{1}{q \alpha^q} y^q_n$. Summing both sides over $n = 1, \ldots, N$ now yields the claim. $\square$
