On the Unintegrated Nevanlinna Fundamental Inequality for Meromorphic Functions of Slow Growth

Abstract

J. Miles proved that for a meromorphic function f and q values a₁, ..., a _q , the inequality $\sum\nolimits_{j = 1}^q {|\frac{{n(r,a_j )}} {{A(r,f)}} - 1| < K}$, holds for some constant K, for all large r in a set of positive lower logarithmic density. This inequality is in some sense stronger than the unintegrated Nevanlinna fundamental inequality $|\sum\nolimits_{j = 1}^q {\frac{{n(r,a_j )}} {{A(r,f)}} - 1| < 2 + o(1)}$. However, it remains the question about the size of the constant K. In this work, the above mentioned inequality will be considered for functions of slow and regular growth, observing that in this case, which is a natural extension of the rational functions class, the constant K can be considerably reduced in relation with the numerical values suggested by Miles. We make use of a result of Barsegian which follows from some beautiful considerations around the main theorems of the Ahlfors theory of covering surfaces.