Lemma 5.24.616E53426F2DB

Proof. By the growth bound, the real translates $h(\cdot +s)$, $s\in [0,1]$, form a family of subharmonic functions that is locally uniformly bounded above. The local upper-envelope theorem therefore shows that

is subharmonic. By periodicity, $H$ is invariant under every real translation, so it depends only on $\operatorname {Im}\zeta $, and its subharmonicity means that it is a convex function of $\operatorname {Im}\zeta $. On the imaginary axis it satisfies $H(\mathrm i y)\leq A\log (2+|y|)+B$. Since a nonconstant convex function on $\mathbb R$ grows at least linearly at one end, this bound shows that $H$ is constant. Thus $h\leq H$ is bounded above, and lemma 5.24.5 shows that $h$ is constant. $\square$