 Solution

A real function $\varphi$ on an interval is a convex function iffor all $a,b$ in its domain and $0\le \theta \le 1$. For a finite-valued random variable $X$ with values $x_j$ and probabilities $p_j$, Jensen's inequality statesApply this with equal probabilities to $a,b$ and the convex function $\varphi (t)=t^p$ on the nonnegative axis:Multiplying by $2^p$ proves the sharp two-term convex power bound. Conversely taking $a=b>0$ forces $2^p\le 2c_p$. ThusFor $p=1$ the inequality is equality for every pair; for $p>1$ equality occurs when $a=b$.