On Integral Functions of Order One and of Finite Type

Let f (z) be an integral function and M(r) the maximum of |f(z)| on |z|=r. The order p and type k of f (z) are defined by the relations

p=lim log logM(r)/log r; k=lim logM(r)/r^p.                                (1)

If p is finite, the function f (z) is said to be of minimal, normal or maximal type according as k vanishes, is a finite positive number, or is infinite. We shall define the number l by the relation

l=lim logM(r)/r^p,                                                                (2)

which might be called the lower type in contrast to k which might be termed the upper type. It is evident that l≤k.