A characterization of Hardy-Orlicz spaces on planar domains

Abstract:

In the paper we prove that for a wide class of bounded domains $D$ in $\mathbb {C}$, a holomorphic function $f$ is in the Hardy-Orlicz space ${H_\phi }(D)$ if and only if \[ \iint _D {\delta (z)\phi ''(\log |f(z)|)\frac {{|f’(z){|^2}}} {{|f(z){|^2}}}dx dy < \infty ,}\] where $\delta (z)$ denotes the distance from $z$ to the boundary of $D$ and $\phi$ is a strongly convex function on $( - \infty ,\infty )$ for which $\phi ''(t)$ exists for all $t$.