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Let

0 for r < 1. Let E o be the Banach space of analytic functions / on the open unit disc D, such that f(z) 0 as z\ -> 1, with norm \\f\\ = svv{\f(z)}E is defined similarly. Let Here, dA denotes two-dimensional Lebesgue measure on D. If geL 1 {D), let [g] denote the coset g+N 1 that contains g. Thus, [g] is an element of the

ABSTRACT. In this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition x + y, multiplication x . y. and two-place exponentiation x '. For a restricted class of such functions, we show that every true identity follows from the natural set of eleven axioms. Thc rates of growth of such functions, in the case of a single independent variable x. as x ~ 00. are also studied. and we give an algorithm for the Hardy relation of eventual domination. again for a restricted class of functions. Value distribution of analytic functions of one and of several complex variables, especially the Nevanlinna characteristic, plays a major role in our proofs.

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