On Maximal Integral Functions

The object of this paper is to discuss the properties of a special class of integral functions defined as follows. Let g(z) be an integral function and p, d two positive numbers. We shall say that g(z) is a maximal integral function of order p and type d when

lim log|g(z)|/|z|^p=d,                          (1)

as |z|→∞ outside a system of circles the sum of whose radii is finite. We shall denote this fact by saying that g(z) belongs to M (P,d).