Convexity gives $(a+b)^p\le 2^{p-1}(a^p+b^p)$ for $p\ge1$.Suggest a slogan
Let \(p\geq1\) and \(a,b\geq0\). Then \[ (a+b)^p \leq 2^{p-1} (a^p + b^p) . \]
The function \(t\mapsto t^p\) is convex on \([0,\infty)\) for \(p\geq1\), so \[ \qty{\frac{a+b}{2}}^p\le\frac{a^p+b^p}{2}. \] Multiplying by \(2^p\) gives the inequality.