Über den Index der Lösungen Linearer Differentialgieichungen

Abstract

In an earliner paper the authors proved the follwoing inequalities for the index (I(r,f) of an entire function f

$$\max (0,\lambda - 1) \leqslant \mathop {\lim }\limits_{r \to \infty } \sup \frac{{l\mathop o\limits^ + g I(r,f)}}{{\log r}} \leqslant \lambda $$

, where λ is the order of f. It is known that the upper bound is sharp. In this paper the authors prove that the lower bound cannot be sharpened if λ≥1 is a rational number. In this direction it is shown that for certain solutions of linear differential equations

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

with polynomial coefficients aj in the left side of the above inequality “≤” and “lim sup” are replaced by “=” and “lim”. Also it is proved that the differential equation has constant coefficients if and only if every solution is of bounded index.