Hayman-Miles Theorem and 2-Valued Algebroid Functions

Abstract

Given an algebroid function w satisfying

$$w^\nu+A_{\nu-1}(z)w^{\nu-1}+\cdot\cdot\cdot+A_0(z)=0,$$

we establish two general methods to calculate the corresponding coefficients in the defining equation for w′ in terms of the meromorphic functions A₀, …, A_ν−1 and their derivatives. We then obtain the following result: Let w be a 2-valued algebroid function. Then T(r,w′) = o(T(r,w)) can hold at most on a set of zero upper logarithmic density.