Zur lokalen Werteverteilung der Lösungen linearer Differentialgleichungen

Abstract

Every solution w of the linear differential equation

$$L_n (w) = w^{(n)} + a_{n - 1^{w^{(n - 1)} } } + \ldots + a_0 w = 0$$

((*))

with polynomial coefficients a_j is a polynomial or an entire function of finite order λ>0. In this paper we prove the following theorem: Let w be a solution of (*) and no polynomial. Let further λ be the order of w and n_a (R, 1/(w−c)) the number of the zeros in the disc |z−a|<R of the function W=w−c. Then there exists a natural number N and a positive constant L such that for every c and |a|>4^1/λ

$$n_a \left( {L|a|^{1 - \lambda } ,1/(w - c)} \right) \leqq N.$$

It is also shown, that for certain solutions of (*) there exists a constant r₀>0 such that we can replace N by n+α for |a|> r₀. α₀ is the degree of the polynomial a₀. An important tool for the proofs is the index of an entire function.