Log-convex sequences and nonzero proximate orders

Abstract

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence $\mathbb{M}=(M_p{)}_{p\in {\mathbb{N}}_0}$, have been put forward by A. Lastra, S. Malek and the second author [1], and their validity depends on the possibility of associating to $\mathbb{M}$ a nonzero proximate order. We provide several characterizations of this and other related properties, in which the concept of regular variation for functions and sequences plays a prominent role. In particular, we show how to construct well-behaved strongly regular sequences from nonzero proximate orders. [1] A. Lastra, S. Malek and J. Sanz, Summability in general Carleman ultraholomorphic classes, J. Math. Anal. Appl. 430 (2015), 1175--1206.

26 pages, this version has been accepted for publication in Journal of Mathematical Analysis and Applications

References in corpus (1)

• Flat functions in Carleman ultraholomorphic classes via proximate orders

Cited by in corpus (7)

• Injectivity and surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes
• Sectorial extensions, via Laplace transforms, in ultraholomorphic classes defined by weight functions
• Indices of O-regular variation for weight functions and weight sequences
• Estimates of formal solutions for some generalized moment partial differential equations
• Summability of formal solutions for a family of generalized moment integro-differential equations
• Solid hulls and cores of classes of weighted entire functions defined in terms of associated weight functions
• Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders