$\mu$ is a (finite) convex function (Phragmén–Lindelöf)

The growth exponent $\mu(\sigma)=\inf\{a\in\mathbb R:\zeta(\sigma+iT)=O(T^{a})\}$ is a real-valued convex function of $\sigma\in\mathbb R$: there is a convex function $f:\mathbb R\to\mathbb R$ with

This is the consequence of the Phragmén–Lindelöf theorem quoted in the source; it is what turns the boundary values $\mu(0)$, $\mu(1)$ into the convexity bound for $\mu(\tfrac12)$.

Formalization Note $\mu$ is LindelofHypothesis.mu, valued in the extended reals. Since $\mu$ is defined in the extended reals, "$\mu$ is a convex function" is stated as: $\mu$ is finite everywhere and coincides with a convex real function.