Optimal tail comparison under convex majorization

Following results of Kemperman and Pinelis, we show that if $X$ and $Y$ are real valued random variables such that $\mathbb{E}\mid Y\mid <\mathrm{\infty }$ and for all non-decreasing convex $\phi :\mathbb{R}\to [0,\mathrm{\infty })$, $\mathbb{E}\phi (X)\le \mathbb{E}\phi (Y)$, then for all $s\in \mathbb{R}$ with $\mathbb{P}\{Y>s\}\mathrm{\ne }0$, $\mathbb{P}\{X\ge \mathbb{E}(Y:Y>s)\}\le \mathbb{P}\{Y>s\}$. This bound is sharp in essentially the strictest possible sense: for any such $Y$ and $s$ there exists such an $X$ with $\mathbb{P}\{X\ge \mathbb{E}(Y:Y>s)\}=\mathbb{P}\{Y>s\}$.