Nicola–Riccardi–Tilli convex-order conjecture for weighted Bergman functions

Let $\mathbb{B}_n$ be the Bergman ball, let $p\in(0,\infty)$ and $\alpha>n$, and let $A^p_\alpha(\mathbb{B}_n)$ denote the weighted Bergman space of holomorphic functions with normalized norm $\|f\|_{A^p_\alpha}=1$. For a convex function $\Phi:[0,1]\to\mathbb{R}$, consider the invariant measure $dv_g$ and the quantity $|f(z)|^p(1-|z|^2)^\alpha$. Nicola–Riccardi–Tilli's conjecture. For every holomorphic function $f\in A^p_\alpha(\mathbb{B}_n)$ normalized by

$$\|f\|_{A^p_\alpha}=1,$$

one has

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

Equivalently, the constant function $f\equiv1$ maximizes this convex functional among normalized holomorphic functions. The conjecture is a convex-order domination principle; the paper establishes only a local validity near $f\equiv1$, so the global assertion remains open.