Given a convex function $Φ:[0,1]\to\mathbb{R}$ and the mean $\mathbb{E}f(\mathbf{X})=a\in[0,1]$, which Boolean function $f$ maximizes the $Φ$-stability $\mathbb{E}[Φ(T_ρf(\mathbf{X}))]$ of $f$? Here $\mathbf{X}$ is a random vector uniformly distributed on the discrete cube $\{-1,1\}^{n}$ and $T_ρ$ is the Bonami-Beckner operator. Special cases of this problem include the (symmetric and asymmetric) $α$-stability problems and the ``Most Informative Boolean Function'' problem. In this paper, we provide several upper bounds for the maximal $Φ$-stability. When specializing $Φ$ to some particular forms, by these upper bounds, we partially resolve Mossel and O'Donnell's conjecture on $α$-stability with $α>2$, Li and Médard's conjecture on $α$-stability with $1<α<2$, and Courtade and Kumar's conjecture on the ``Most Informative Boolean Function'' which corresponds to a conjecture on $α$-stability with $α=1$. 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.