Sur une in\'egalit\'e fonctionnelle sur les vari\'et\'es toriques avec application \`a la torsion analytique holomorphe

Let $X$ be a complex non-singular toric variety, $E$ an equivariant and ample line bundle on $X$. We introduce a new functional $V_{\mathrm{\infty }}$ on the set of smooth, positive and invariant metrics on $E$. We compare $V_{\mathrm{\infty }}$ to some classical functionals. As an application, we study the variation of the holomorphic analytic torsion.