On the Phragmen Lindelöf Theorem in strips

In this paper we study an $L^p$ analogue of Bohr’s abscissae of summability for Dirichlet series. For polynomially bounded analytic functions in a strip with order function $\mu$, convexity of $1\mathrm{/}\mu$ is equivalent to approximate concavity of the abscissae in $p$. If $\mu$ obeys a functional equation of the Selberg class type, this is equivalent to the Lindelöf hypothesis if ${\mu }^{\mathrm{\prime }}(1\mathrm{/}2)$ does not exist. Otherwise, $\mu$ is everywhere differentiable (therefore subconvex) with quadratic decay near one.