A class of functions in F whose integral is not in F .

摘要

For a function f, meromorphic in the complex plane, R. Nevanlinna noted that its characteristic function T( r, f) could be used to categorize f according to its rate of growth as |z| = r approached infinity. If f is a meromorphic function in the unit disk D = {z : |z| < 1}, many value distribution results which are analogous to those for functions defined in the plane may be proved provided limsupr→1- Tr,f -log1-r =+infinity. If class F is defined to be those functions meromorphic in D for which limsupr→1- Tr,f -log1-r =af<+infinity, then D. Shea and L. Sons[19] developed properties of functions in F and showed F is not closed under integration. In this dissertation we consider the class S of analytic functions in D which are in F , but have integrals not in F . We explore characteristics of the class and show some examples. We explore the manner in which these functions behave in comparison with other function classes in the unit disk D. We consider the power series representation for such functions in S and look at what conditions must be imposed on the coefficients of a power series about zero. We also examine S in terms of value distribution, proving some results about the number of zeros of functions f ∈ S as well as those of their integrals. Finally we consider various representations for functions in S , including gap series and Tsuji products.