Li–Solovej convex integral inequality for holomorphic functions on the complex ball

Let ${\mathbb B}_n$ be the unit ball in $\mathbb C^n$, with Lebesgue measure $dv(z)$, and define the hyperbolic measure by

$$dv_g(z)=\frac{dv(z)}{(1-|z|^2)^{n+1}}.$$

For $0<p<\infty$ and $\alpha>n$, let $A^p_\alpha$ be the space of holomorphic functions $f$ on ${\mathbb B}_n$ satisfying

$$\|f\|^p_{A^p_\alpha}:=c_n\int_{{\mathbb B}_n}|f(z)|^p(1-|z|^2)^\alpha\,dv_g(z)<\infty,$$

where $c_\alpha=\frac{\Gamma(\alpha)}{\alpha!\Gamma(\alpha-n)}$, and hence $\|1\|_{A^p_\alpha}=1$. Li–Solovej's conjecture. For every convex function $\Phi:[0,1]\to\mathbb R$, every $p\in(0,\infty)$, every $\alpha>n$, and every $f\in A^p_\alpha$ with $\|f\|_{A^p_\alpha}=1$,

$$\int_{{\mathbb B}_n}\Phi\bigl(|f(z)|^p(1-|z|^2)^\alpha\bigr)\,dv_g(z)\leq\int_{{\mathbb B}_n}\Phi\bigl((1-|z|^2)^\alpha\bigr)\,dv_g(z).$$

The inequality was conjectured in dimension $n=1$ and was proved there, with a characterization of extremizers. It remains open in dimensions $n\geq2$; the isoperimetric conjecture that isoperimetric subsets of ${\mathbb B}_n$ are geodesic balls is known to imply it.