通过近似阶的Phragmén-Lindelöf定理和渐近性的传播
A Phragmén-Lindelöf theorem via proximate orders, and the propagation of asymptotics
摘要：
证明了对于$\mathbb{C}$中某个扇区内的渐近有界全纯函数，若在一个顶点方向上的渐近展开，且其约束条件为一个允许非零近似阶的渐近凸序列，则其在整个扇区内的渐近展开，且其控制条件为同一序列.这推广了A. Fruchard和C.张为Gevrey渐近展开，并证明强烈依赖于一个适当的改进版本的经典Phragmén-Lindelöf定理，这里获得的功能，其增长在一个部门指定的意义上的非零近似秩序E。Lindelöf和G. Valiron。
英文摘要：
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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