# A Phragm\'en-Lindel\"of theorem via proximate orders, and the propagation of asymptotics

## TL;DR

This paper extends the Phragmén-Lindelöf theorem to functions with growth controlled by proximate orders, showing that asymptotic expansions in one direction imply expansions in entire sectors under certain conditions.

## Contribution

It generalizes existing results by incorporating proximate orders into the Phragmén-Lindelöf framework for asymptotic analysis.

## Key findings

- Proves a generalized Phragmén-Lindelöf theorem for proximate orders.
- Shows asymptotic expansions in a single direction extend to entire sectors.
- Provides conditions under which asymptotic control propagates across sectors.

## Abstract

We prove that, for asymptotically bounded holomorphic functions in a sector in $\mathbb{C}$, an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by A. Fruchard and C. Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragm\'en-Lindel\"of theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of E. Lindel\"of and G. Valiron.