Sharp power-call inequality ($s^{1+\epsilon }\ge \frac{(1+\epsilon {)}^{1+\epsilon }}{{\epsilon }^{\epsilon }}{K}^{\epsilon }(s-K{)}_{+}$)

For $K>0$ and $s>K$, minimize $(s/K{)}^{1+\epsilon }/(s/K-1)$. Its minimum occurs at $s/K=(1+\epsilon )/\epsilon$, giving the sharp constant. Taking expected values yields a uniform bound on $K^{\epsilon }C(K)$ from a finite moment of order $1+\epsilon$.