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Abstract

Let $I, J\subset \mathbb{R}$ be closed intervals, and let $H$ be a $C^{3}$ 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 the Lebesgue measure. We show that for the inequality $$ \int_{\mathbb{R}^{n}} \operatorname{ess\,sup}_{y \in \mathbb{R}^{n}} H\left( f\left(\frac{x-y}{a}\right),g \left(\frac{y}{b}\right)\!\right)\,d\mu (x) \geq H\Bigl(\int_{\mathbb{R}^{n}}f\,d\mu, \int_{\mathbb{R}^{n}}g\,d\mu \Bigr) $$ to hold for all Borel functions $f,g$ with values in $I$ and $J$ respectively, it is necessary that \begin{gather*} 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,\quad\ a+b\geq 1\quad \text{and}\quad \int_{\mathbb{R}^{n}}x\,d\mu=0\quad \text{if $a+b \gt 1$.} \end{gather*} Moreover, if $d\mu$ is a Gaussian measure then the necessary condition becomes sufficient. This extends Prékopa–Leindler and Ehrhard inequalities to an arbitrary function $H(x,y)$. As an immediate application we obtain a new proof of the Ehrhard inequality. In particular, we show that in the class of even probability measures with smooth positive density and finite moments the Gaussian measure is the only one which satisfies the functional form of the Ehrhard inequality on the real line with its own distribution function.