ON THE PROXIMATE ORDER WITH RESPECT TO THE MODEL FUNCTION

We investigate the generalization of the proximate order in the Valiron sense. 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 growth estimate of the function with respect to the model function. We discuss the generalized proximate order corresponding to an arbitrary model function of growth. We also consider some properties of the generalized proximate order in the case when the model function of growth is multiplicative. In particular, we prove that the smoothness conditions on the proximate order do not matter.