The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem.

Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [0,1]$. Let $X$ and $Y$ be standard Gaussians with covariance $\rho$. Prove that $\mathbf{E}[f(X)f(Y)]$ ≤ $\mathbf{E}[f(X)^2 \mathrm{sgn}(X) \mathrm{sgn}(Y)]$.

The quantity on the left-hand side arises naturally in many contexts; e.g., it is the integral of $f(x)f(y)$ against the Mehler kernel (with parameter $\rho$).

I have some reason to believe this inequality is true. For one piece of evidence, suppose $f$'s range is $\pm 1$. Then the inequality reduces to

$\mathbf{E}[f(X)f(Y)]$ ≤ $\mathbf{E}[\mathrm{sgn}(X) \mathrm{sgn}(Y)]$;

i.e., it's saying that $\mathrm{sgn}$ is the $\pm 1$-valued odd function maximizing the LHS. This is indeed true; it follows from a result of Christer Borell ("Geometric bounds on the Ornstein-Uhlenbeck velocity process"), proved by Ehrhard symmetrization. It was also given a different proof by Beckner, deducing it from a rearrangment inequality on the sphere.

The second inequality generalizes to the case of functions $f : \mathbb{R}^n \to$ {$-1,1$}, but I believe the first inequality, which I would like to prove, is inherently $1$-dimensional.

Any ideas, or pointers to literature that might help? Thanks!