Power inequality for symmetric Markov energies ($\mathcal{E}(f^{q/2})\le \frac{q^2}{4(q-1)}\mathcal{E}(f^{q-1},f)$)

For $a,b\ge 0$ and $q>2$, Cauchy-Schwarz inequality applied to the integral of $(q/2)s^{q/2-1}$ gives $(a^{q/2}-b^{q/2}{)}^2\le q^2(a^{q-1}-b^{q-1})(a-b)/(4(q-1))$. Integrating against the symmetric two-point kernel measure and taking the energy limit proves the displayed inequality.

// Target: analysis.bigb