Characterizations of Hardy–Orlicz and Bergman–Orlicz spaces

Let $\widetilde\nabla$ и $\tau$ denote the invariant gradient and invariant measure on the unit ball $B$ of $\mathbb C^n$, respectively. Assume that $f$ is a holomorphic function on $B$ and $\varphi\in C^2 ({\mathbb R})$ is a nonnegative nondecreasing convex function. Then $f$ is in the Hardy–Orlicz space $H_\varphi(B)$ if and only if
$$ \int_B\varphi''(\log|f(z)|)\frac{|\widetilde\nabla f(z)|^2}{|f(z)|^2}(1-|z|^2)^n\,d\tau(z)<\infty. $$
Analogous characterizations of Bergman–Orlicz spaces are obtained.