On the Phragmen Lindel\"of Theorem in strips

Abstract

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 $1/\mu$ 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 Lindel\"of hypothesis if $\mu'(1/2)$ does not exist. Otherwise, $\mu$ is everywhere differentiable (therefore subconvex) with quadratic decay near one.