 Sharp two-term convex power bound

Apply Jensen's inequality to the two equally weighted values $a,b$ and the convex function $t\mapsto t^p$. Equality for $a=b>0$ proves that $2^{p-1}$ is the smallest possible constant. For $p>1$, strict convexity makes equality equivalent to $a=b$; for $p=1$, equality holds for every pair.