Sharp Logarithmic Derivative Estimates With Applications to Ordinary Differential Equations in the Unit Disc

Abstract

New estimates are obtained for the maximum modulus of the generalized logarithmic derivatives, 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.2000 Mathematics subject classification: primary 34M10; secondary 30D30.

Finiteness of φ-order of solutions of linear differential equations in the unit disc

“…Conditions on the coefficients forcing the solutions to belong to certain function spaces can be found in [14,19,27,28]. The cases where the solutions either belong to the Nevanlinna class or are non-admissible are studied in [4,14,18,27], while results on (1.1) with solutions of finite order can be found in [4,6,8,9,14,18,21,22]. Fast growing solutions have been studied in terms of the iterated order of growth in [5,17].…”

Wiman-Valiron theory for a class of functions meromorphic in the unit disc

“…Theorem 1.1 Let the function f be meromorphic in the unit disc D with a direct tract. Using the notation (3), (4) and (5) set ε(r) = min 1 − r 2a(r) β (log a(r)) 1+δ , 1 a(r) 1−β (log a(r)) 1+δ (6) for r 0 ≤ r < 1. Then there exists a set E ⊆ [r 0 , 1) satisfying…”

On the Minimum Modulus of Analytic Functions of Moderate Growth in the Unit Disc

“…Applications of this concept to factorization of analytic functions in D, and logarithmic derivative estimates can be found in [6,9]. Let a sequence (a n ) in D satisfy the condition…”