Sharp two-term convex power bound
= Sharp two-term convex power bound
{title2=$(a+b)^p\le2^{p-1}(a^p+b^p),\quad a,b\ge0,\ p\ge1$}
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.