PomeCorollary 5.41.2.B98B62E1E7D5 Let $I\subset \mathbb R$ be a nonempty open interval. A radial function $z\mapsto q(\log |z|^2)$ is subharmonic on the annulus $\{ z:\log |z|^2\in I\} $ if and only if $q$ is finite and convex on $I$. If the function identically equal to $-\infty $ is also regarded as subharmonic, the alternative $q\equiv -\infty $ has to be added.
Corollary 5.41.2B98B62E1E7D5
Proof. The holomorphic covering map $z=\mathrm{e}^{w/2}$ pulls the function back to $q(\operatorname{Re}w)$, and the argument of lemma 5.41.1 proves necessity. Conversely, a finite convex function $q$ has nonnegative distributional second derivative. Thus $q(\operatorname{Re}w)$ is subharmonic, and subharmonicity descends through the local biholomorphisms of the covering. $\square$
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