Given
$a, \,b \,\in\, \mathbb{R}_{\gt 0}$, non-negative real numbers;
$p, \, q \,\in\, \mathbb{R}_{\gt 1}$ such that
then the following inequality holds:
which is an equality if and only if $a^p = b^q$.
One proof is by convexity of the exponential function: choosing $x, y, t$ such that $\exp(x) = a^p$, $\exp = b^q$ and $t = \frac1{p}$, Young’s inequality is identical to the convexity constraint
See also:
Last revised on December 27, 2022 at 19:10:59. See the history of this page for a list of all contributions to it.