The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc

Abstract

In this paper, we give the definition of iterated order to classify functions of fast growth in the unit disc, and investigate the growth of solutions of linear differential equations with analytic coefficients of iterated order in the unit disc. We obtain several results concerning the iterated order of solutions.  2005 Elsevier Inc. All rights reserved.

Linear Differential Equations with Solutions in the Dirichlet Type Subspace of the Hardy Space

“…As for local considerations, Nevanlinna theory has been applied to fast growing analytic solutions [3], [4], [7], [8], [16], [18], [21], [22], but the analysis of slowly growing solutions seems to require a different approach [16], [17], [19], [26], [30]. Chr.…”

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

“…In the paper [6], Cao and Yi studied the properties of solutions to the arbitrary order linear differential equations in of the form…”

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

“…Now we give the definitions of iterated order and growth index to classify generally the functions of fast growth in ∆ as those in C (see [4] , [17] , [18]) . Let us define inductively, for r ∈ [0, 1) , exp 1 r = e r and exp p+1 r = exp exp p r , p ∈ N. We also define for all r sufficiently large in (0, 1) , log 1 r = log r and log p+1 r = log log p r , p ∈ N. Moreover, we denote by exp 0 r = r, log 0 r = r, exp −1 r = log 1 r, log −1 r = exp 1 r. Definition 1.3 [5,6] The iterated p−order of a meromorphic function f in ∆ is defined by…”

“…Definition 1.4 [5] The growth index of the iterated order of a meromorphic function f (z) in ∆ is defined by…”