Variations on the proximate order

TL;DR

This paper offers a new, generalized interpretation of the concept of proximate order in complex analysis, simplifying its definition and extending its applicability to various classes of functions.

Contribution

It introduces a unified, minimal-condition framework for proximate order, broadening its theoretical foundation and providing a new perspective on classical concepts.

Findings

Provides a general interpretation as a proximate growth function

Simplifies the definition with a single condition

Extends the concept to various types of functions

Abstract

The concept of proximate order is widely used in the theories of entire, meromorphic, subharmonic and plurisubharmonic functions. We give a general interpretation of this concept as a proximate growth function relative to a model growth function. If a function is the proximate growth function with respect to the identity function on the positive semi-axis, then the logarithm of this function is the classical proximate order. Our definition uses only one condition. This form of definition is also new for the classical proximate order.