Meromorphic Functions of Finite Order

Abstract

In the preceding discussion we have frequently observed that the degree of transcendency of a meromorphic function is reflected in the growth of its characteristic T(r). Considered as a class, meromorphic functions whose characteristic is of a definite order of magnitude have a more complicated structure the higher the order of growth. Thus, for instance, we have found that among other functions meromorphic for |z| < 1, meromorphic functions which have a bounded characteristic in the unit disk are distinguished by certain simple properties. They have, for example, definite boundary values almost everywhere on the bounding circle |z| = 1, which is not generally the case otherwise. The boundedness of the characteristic of a function meromorphic in the entire plane z ≠ ∞ implies that the function is constant. For rational functions T(r) = O(log r), while for a transcendental function the ratio T(r):log r is unbounded as r → ∞¹.