On the $\mathrm{\Phi }$-Stability and Related Conjectures

Given a convex function $\mathrm{\Phi }:[0,1]\to \mathbb{R}$ and the mean $\mathbb{E}f(\mathbf{X})=a\in [0,1]$, which Boolean function $f$ maximizes the $\mathrm{\Phi }$-stability $\mathbb{E}[\mathrm{\Phi }(T_{\rho }f(\mathbf{X}))]$ of $f$? Here $\mathbf{X}$ is a random vector uniformly distributed on the discrete cube $\{-1,1{\}}^n$ and $T_{\rho }$ is the Bonami-Beckner operator. Special cases of this problem include the (symmetric and asymmetric) $\alpha$-stability problems and the Most Informative Boolean Function'' problem. In this paper, we provide several upper bounds for the maximal $\Phi$-stability. When specializing $\Phi$ to some particular forms, by these upper bounds, we partially resolve Mossel and O'Donnell's conjecture on $\alpha$-stability with $\alpha>2$, Li and M\'edard's conjecture on $\alpha$-stability with $1<\alpha<2$, and Courtade and Kumar's conjecture on the Most Informative Boolean Function’’ which corresponds to a conjecture on $\alpha$-stability with $\alpha =1$. Our proofs are based on discrete Fourier analysis, optimization theory, and improvements of the Friedgut–Kalai–Naor (FKN) theorem. Our improvements of the FKN theorem are sharp or asymptotically sharp for certain cases.