Let $I, J\subset \mathbb{R}$ be closed intervals, and let $H$ be $C^{3}$ smooth real valued function on $I\times J$ with nonvanishing $H_{x}$ and $H_{y}$. Take any fixed positive numbers $a,b$, and let $d\mu$ be a probability measure with finite moments and absolutely continuous with respect to Lebesgue measure. We show that for the inequality $$ \int_{\mathbb{R}^{n}} \mathrm{ess\,sup}_{y \in \mathbb{R}^{n}}\; H\left( f\left(\frac{x-y}{a}\right),g\left(\frac{y}{b}\right)\right)d\mu (x) \geq H\left(\int_{\mathbb{R}^{n}}fd\mu, \int_{\mathbb{R}^{n}}gd\mu \right) $$ to hold for all Borel functions $f,g$ with values in $I$ and $J$ correspondingly it is necessary that $$ a^{2}\frac{H_{xx}}{H_{x}^{2}}+(1-a^{2}-b^{2})\frac{H_{xy}}{H_{x}H_{y}}+b^{2}\frac{H_{yy}}{H_{y}^{2}}\geq 0, $$ $|a-b|\leq 1$, $a+b\geq 1$ and $\int_{\mathbb{R}^{n}}xd\mu=0$ if $a+b>1$. Moreover, if $d\mu$ is a Gaussian measure then the necessary condition becomes sufficient. This extends Pr\'ekopa--Leindler and Ehrhard in 查看全文>>