Log-convex sequences and nonzero proximate orders

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence M=(Mp)p∈N0, have been put forward by A. Lastra, S. Malek and the second author [10], and their validity depends on the possibility of associating to 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.