The sharp constant in the Chang--Marshall inequality

We prove that the optimal constant in the Chang–Marshall inequality is $e$, as conjectured by Andreev and Matheson. More precisely, if $f$ is analytic in the unit disk, vanishes to order $m\ge 1$ at the origin, and has normalized Dirichlet integral $d\le 1$, then the boundary mean of $e^{\mathrm{\mid }f{\mathrm{\mid }}^2}$ is at most $e^{d\mathrm{/}m}$. Equality holds precisely for monomials. The proof uses an auxiliary closed planar curve and the isoperimetric inequality to establish monotonicity of the exponential functional along normalized radial dilations.