FINITENESS OF φ-ORDER OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS IN THE UNIT DISC

If phi : [0, 1) -> (0, infinity) is a non- decreasing unbounded function, then the phi-order of a meromorphic function f in the unit disc is defined as sigma phi(f) = lims up log+T(r, f)/log phi(r)r -> 1(-) where T (r, f) is the Nevanlinna characteristic of f. In particular, sigma 1/1-r (f) order of f, and sigma log 1/1-r (f) is the logarithmic order of f. Several results on the finiteness of the phi-order of solutions of f((k)) + A(k-1)(z) f((k-1)) + ... + A(1)(z) f' + A(0)(z) f = 0 are obtained in the case when the coefficients A(0)(z), ... , A(k-1)(z) are analytic functions in the unit disc. This paper completes some earlier results by various authors.