= Solution
A real function $\phi$ on an interval is a <convex function> if
$$
\phi(\theta a+(1-\theta)b)\le\theta\phi(a)+(1-\theta)\phi(b)
$$
for all $a,b$ in its domain and $0\le\theta\le1$. For a finite-valued <random variable> $X$ with values $x_j$ and probabilities $p_j$, <Jensen's inequality> states
$$
\phi\left(\sum_jp_jx_j\right)\le\sum_jp_j\phi(x_j),
\qquad\text{equivalently }\phi(\mathbb EX)\le\mathbb E\phi(X).
$$
Apply this with equal probabilities to $a,b$ and the <convex function> $\phi(t)=t^p$ on the nonnegative axis:
$$
\left(\frac{a+b}{2}\right)^p\le\frac{a^p+b^p}{2}.
$$
Multiplying by $2^p$ proves the <sharp two-term convex power bound>. Conversely taking $a=b>0$ forces $2^p\le2c_p$. Thus
$$
\boxed{c_p=2^{p-1}.}
$$
For $p=1$ the inequality is equality for every pair; for $p>1$ equality occurs when $a=b$.
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