On the proximate growth function relative to the model growth function

The concept of proximate order is widely used in the theories of integer, meromorphic, subharmonic, and plurisubharmonic functions. In this paper, we provide a general interpretation of this concept as a proximate growth function relative to the model growth function. The classical proximate order is the proximate order in the sense of Valiron. Our definition uses only one condition. This form of definition is new for the classical proximate order. In this review, we show that for any function defined on a positive ray whose growth is determined by a model growth function, there is a proximate growth function relative to the model growth function.