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A holomorphic function on the unit disc has bounded characteristic when ${\sup }_{r<1}\int {\log }^{+}|f(r\xi )|dm(\xi )<\infty$, where ${\log }^{+}t=\max (0,\log t)$. Thus it belongs to the Nevanlinna class. A general meromorphic function requires the additional pole-counting term in its characteristic; the displayed analytic definition has no such term.

Use the absolute value around the logarithm in the original PDF. Set $v(z)={\log }^{+}|f(z)|=\max (0,\log |f(z)|)$. This is a nonnegative subharmonic function, including at zeros by its continuous extension. Its radial means are nondecreasing: Poisson integral comparison on a larger concentric circle bounds the smaller mean by the larger one. The hypothesis givesConsequently $T(r)=0$ for every $r$, and continuity of $v$ implies $v=0$ throughout the unit disc. Thus $|f|\le 1$ everywhere, which is stronger than the requested conclusion. In particular $f$ has bounded characteristic, because ${\sup }_{r}T(r)<\infty$; it belongs to the Nevanlinna class and to H-infinity space. The hypothesis also excludes the identically zero function, whose absolute logarithmic integral is infinite.