Fast Growing Solutions of Linear Differential Equations in the Unit Disc

Abstract

For n ∈ N, the n-order of an analytic function f in the unit disc D is defined bywhere log + x = max{log x, 0}, log + 1 x = log + x, log + n+1 x = log + log + n x, and M (r, f ) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equationwhere the coefficients are analytic in D, satisfy σM,n+1(f ) ≤ α if and only if σM,n(aj) ≤ α for all j = 0, . . . , k − 1. Moreover, if q ∈ {0, . . . , k − 1} is the largest index for which σM,n(aq) = max 0≤j≤k−1 σM,n(aj) , then there are at least k − q linearly independent solutions f of ( †) such that σM,n+1(f ) = σM,n(aq).Some refinements of these results in terms of the n-type of an analytic function in D are also given. Mathematics Subject Classification (2000). Primary 34M10; Secondary 30D35.

Value distributions of solutions to complex linear differential equations in angular domains

“…In this article, we assume the reader is familiar with standard notations and basic results of Nevanlinna's value distribution theory in the unit disk D fz W jzj < 1g, in an angular region, and in the complex plane C respectively; see [1][2][3][4][5]. The order .f / and lower order .f / of f which is meromorphic in C or are defined as follows: …”

“…; kg < p; .A 0 /, then i.f / D p C 1 and p; .A 0 / Ä pC1; .f / holds for all solutions f 6 Á 0 of equation (2).…”

Properties of higher order differential polynomials generated by solutions of complex differential equations in the unit disc

“…Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna's value distribution theory on the complex plane and in the unit disc ∆ = {z : |z| < 1} (see [13] , [14] , [18] , [20]). We need to give some definitions and discussions.…”

“…Definition 1.5 [3,15,20] The iterated p−type of a meromorphic function f of iterated p−order ρ (0 < ρ < ∞) in ∆ is defined by…”

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

“…Sharp growth estimates for the maximum modulus of the generalized logarithmic derivative f (k) / f ( j) , where f is meromorphic in D and k and j are integers satisfying k > j ≥ 0, are obtained in [3,8]. The special cases where f either belongs to the Nevanlinna class or is a Blaschke product are further discussed in [9].…”