On the counting function for the 𝑎-values of a meromorphic function.

Abstract

Introduction. If f(z) is a nonconstant meromorphic function in IzI <cxc, we let n(r, a) denote the number of roots counting multiplicities of the equation f(z) =a in IzI < r. Our principal result is an unintegrated analogue for n(r, a) of the theorem which asserts that the Valiron deficient values of f(z) have inner capacity zero. Our result contains both an exceptional set of a-values and an exceptional set of r-values. We also obtain a result on supa n(r, a) having an exceptional set of a-values which bears on a question of Hayman and Stewart. We show by examples that all the exceptional sets in our results are in general nonempty. One of our examples also shows that the exceptional set of r-values in Ahlfors' theory of covering surfaces is in general nonempty.