A Phragmén–Lindelöf Theorem via Proximate Orders, and the Propagation of Asymptotics

Abstract

We prove that, for asymptotically bounded holomorphic functions in a sector in 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.AMS Classification: 30E15, 30C80, 26A12, 30H50.We warn the reader that there is no agreement about the terminology in this respect: while most authors adhere, as we will do, to the convention that the asymptotics in (1.1) is Gevrey of order 1/k, others (for example, Fruchard and Zhang or W. Balser in [1]) say this is of order k. Moreover, the notion of type is not standard, compare with the definition by M. Canalis-Durand [2] for whom the type in case one has (1.1) is (1/R + δ) k . It should also be mentioned that the factor Γ(1 + p/k) could be changed into (p!) 1/k without changing the asymptotics, but this would affect the base of the geometric factor providing the type (by Stirling's formula, see [2, pp. 3-4]) in any case. As it will be explained below, our interest in the type will be limited, and so we will choose a simple approach in this respect, see Definitions 2.2 and 2.11.The proof of this result is based on the classical Phragmén-Lindelöf theorem and on the so-called Borel-Ritt-Gevrey theorem. This last statement provides the surjectivity, as long as the opening of the sector is at most π/k, of the Borel map sending a function with Gevrey asymptotic expansion of order 1/k in a sector to its series of asymptotic expansion, whose coefficients will necessarily satisfy Gevrey-like estimates. Also, the injectivity of the Borel map in sectors of opening greater than π/k (known as Watson's lemma) plays an important role when specifying conditions that guarantee the uniqueness of a function with a prescribed Gevrey asymptotic expansion of order 1/k in a direction.The main aim of this paper is to extend these results for other types of asymptotic expansions available in the literature. This possibility was already mentioned in [11], where A. Lastra, J. Mozo-Fernández and the second author of this paper generalized the results of Fruchard and Zhang for holomorphic functions of several variables in a polysector (cartesian product of sectors) admitting strong asymptotic expansion in the sense of H. Majima [12,13], considering also the Gevrey case as introduced by Y. Haraoka in [5].The asymptotics we will consider are those associated to the consideration of general ultraholomorphic classes of functions defined by constraining the growth of the sequence of their successive derivatives in a sector in terms of a sequence M = (M p ) p∈N0 of positive numbers (N 0 ...