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arXiv · 1706.08804

A Phragmén-Lindelöf theorem via proximate orders, and the propagation of asymptotics

Javier Jiménez-Garrido·Javier Sanz·Gerhard Schindl
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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 Phragmén-Lindelöf theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of E. Lindelöf and G. Valiron.

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Javier Jiménez-Garrido, Javier Sanz, Gerhard Schindl. 2017-06-27. A Phragmén-Lindelöf theorem via proximate orders, and the propagation of asymptotics. https://doi.org/10.1007/s12220-019-00203-5%2C

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