Proposition 17.3.9356D15D0C316

Proof. Let $u_ t=(1-t)u+tv$ and $h=v-u$. Then $\partial _ t\mu _{u_ t}=n\, \mathrm{dd}^{\mathrm{c}}h\wedge \langle \theta _{u_ t}^{n-1}\rangle $. Indeed, by multilinearity and (17.2.1), both sides expand into combinations of the mixed measures $\langle \theta _ u^ a\wedge \theta _ v^{n-a}\rangle $, obtained by substituting $\langle \theta _ u^ a\wedge \theta _ v^{n-a}\rangle $ for $y^ az^{n-a}$ in the two sides of the identity $\partial _ t((1-t)y+tz)^ n=n(z-y)((1-t)y+tz)^{n-1}$ in commuting variables $y,z$. Differentiating (17.3.5) once more, again as a polynomial in $t$, and using this identity, we obtain

by lemma 17.2.7. Thus $t\mapsto E_\theta ^\phi (u_ t)$ is concave, and its derivative decreases from $\int _ Xh\, \mu _ u$ at $t=0$ to $\int _ Xh\, \mu _ v$ at $t=1$. Integrating the derivative over $[0,1]$ gives the tangent bounds (17.3.10). Since $\int _ X\langle \theta _{u_ t}^ n\rangle =m$, the first variation formula also gives $|E_\theta ^\phi (v)-E_\theta ^\phi (u)|\leq m\sup |v-u|$. Finally, the cocycle formula (17.3.6) applied to the pair $\psi ,u$ reads $E_\theta ^\phi (u)-E_\theta ^\phi (\psi )=E_\theta ^\psi (u)$. $\square$