SHARP LOGARITHMIC DERIVATIVE ESTIMATES WITH APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS IN THE UNIT DISC

Abstract

Abstract New estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f ( k ) / f ( j ) , where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k > j ≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.