Generalization of proximate order and applications

TL;DR

This paper introduces the concept of quasi proximate order, a generalization of proximate order, to analyze analytic functions with differing order and lower order of growth, and proves an existence theorem for it.

Contribution

It develops the theory of quasi proximate order and extends classical results on the asymptotic behavior of analytic functions in the unit disc.

Findings

Established the existence of quasi proximate order functions.

Generalized Valiron's theorem for quasi proximate orders.

Extended results on asymptotic behavior of analytic functions.

Abstract

We introduce a concept of a quasi proximate order which is a generalization of a proximate order and allows us to study efficiently analytic functions whose order and lower order of growth are different. We prove an existence theorem of a quasi proximate order, i.e. a counterpart of Valiron's theorem for a proximate order. As applications, we generalize and complement some results of M. Cartwright and C.~N.~Lin\-den on asymptotic behavior of analytic functions in the unit disc.