Log-convex sequences and nonzero proximate orders

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.