= Solution
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 gives
$$
0\le T(r)=\int\log^+|f(re^{i\theta})|\,dm
\le\int\big|\log|f(re^{i\theta})|\big|\,dm\longrightarrow0.
$$
Consequently $T(r)=0$ for every $r$, and continuity of $v$ implies $v=0$ throughout the <unit disc>. Thus \b[$|f|\le1$ everywhere], which is stronger than the requested conclusion. In particular $f$ has <bounded characteristic>, because $\sup_rT(r)<\infty$; 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.
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