On the $$\Phi $$-stability and related conjectures

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