Power inequality for symmetric Markov energies
= Power inequality for symmetric Markov energies
{title2=$\mathcal E(f^{q/2})\leq\frac{q^2}{4(q-1)}\mathcal E(f^{q-1},f)$}
For $a,b\geq0$ and $q>2$, <Cauchy-Schwarz inequality> applied to the integral of $(q/2)s^{q/2-1}$ gives $(a^{q/2}-b^{q/2})^2\leq q^2( a^{q-1}-b^{q-1})(a-b)/(4(q-1))$. Integrating against the symmetric two-point kernel measure and taking the energy limit proves the displayed inequality.
// Target: analysis.bigb