Given a convex function $\Phi:[0,1]\to\mathbb{R}$ 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 $f$? 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.

翻译：给定凸函数$\Phi:[0,1]\to\mathbb{R}$与均值$\mathbb{E}f(\mathbf{X})=a\in[0,1]$，何种布尔函数$f$能最大化$f$的$Φ$-稳定性$\mathbb{E}[\Phi(T_{\rho}f(\mathbf{X}))]$？此处$\mathbf{X}$为均匀分布于离散立方体$\{-1,1\}^{n}$上的随机向量，$T_{\rho}$为Bonami-Beckner算子。该问题的特例包括（对称与非对称）$\alpha$-稳定性问题及“最信息性布尔函数”问题。本文给出了最大$Φ$-稳定性的数个上界。当将$Φ$特化为特定形式时，借助这些上界，我们部分解决了Mossel与O'Donnell关于$\alpha>2$时$\alpha$-稳定性的猜想、Li与Médard关于$1<\alpha<2$时$\alpha$-稳定性的猜想，以及Courtade与Kumar关于“最信息性布尔函数”的猜想（该猜想对应$\alpha=1$时的$\alpha$-稳定性问题）。我们的证明基于离散傅里叶分析、优化理论以及对Friedgut-Kalai-Naor（FKN）定理的改进。本文对FKN定理的改进在某些情形下是精确或渐近精确的。