Sharp power-call inequality
= Sharp power-call inequality
{title2=$s^{1+\varepsilon}\ge\frac{(1+\varepsilon)^{1+\varepsilon}}{\varepsilon^\varepsilon}K^\varepsilon(s-K)_+$}
For $K>0$ and $s>K$, minimize $(s/K)^{1+\varepsilon}/(s/K-1)$. Its minimum occurs at $s/K=(1+\varepsilon)/\varepsilon$, giving the sharp constant. Taking <expected values> yields a uniform bound on $K^\varepsilon C(K)$ from a finite moment of order $1+\varepsilon$.