About global properties of proximate order

Proximate orde is important in the theory of subharmonic and -subharmonic functions. Classical properties were presented in many monographs, for example in [1]. Note that using the proximate order of A.F. Grishin studied the growth of subharmonic and -subharmonic functions at infinity. In the article we sharpen Grishin’s variant of the property theorems of proximate order. The result of this paper allows us to somewhat simplify the constructions from the proof of several assertions.