Let $\mathbf{X}$ be a random variable uniformly distributed on the discrete cube $\{ -1,1\} ^{n}$, and let $T_{\rho}$ be the noise operator acting on Boolean functions $f:\{ -1,1\} ^{n}\to\{ 0,1\} $, where $\rho\in[0,1]$ is the noise parameter, representing the correlation coefficient between each coordination of $\mathbf{X}$ and its noise-corrupted version. Given a convex function $\Phi$ and the mean $\mathbb{E}f(\mathbf{X})=a\in[0,1]$, which Boolean function $f$ maximizes the $\Phi$-stability $\mathbb{E}[\Phi(T_{\rho}f(\mathbf{X}))]$ of $f$? 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.
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