Strongly regular sequences and proximate orders

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence M=(Mp)p∈N0M=(Mp)p∈N0, have been put forward by A. Lastra, S. Malek and the second author [28]. We study several open questions related to the existence of kernels of summability constructed by means of analytic proximate orders. In particular, we give a simple condition that allows us to associate a proximate order with a strongly regular sequence. Under this assumption, and through the characterization of strongly regular sequences in terms of so-called regular variation, we show that the growth index γ(M)γ(M) defined by V. Thilliez [54] and the order of quasianalyticity ω(M)ω(M) introduced by the second author [50] are the same.