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

Abstract

If ϕ : [0, 1) → (0, ∞) is a non-decreasing unbounded function, then the ϕ-order of a meromorphic function f in the unit disc is defined asthe order of f , and σ log 1 1−r (f ) is the logarithmic order of f . Several results on the finiteness of the ϕ-order of solutions ofare 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.

On solutions of linear differential equations of arbitrary fast growth in the unit disc

“…Our results do not intersect with that from [4]. The following theorem generalize Theorem D and is a counterpart of a result from [6] proved for entire functions.…”

“…There has been an increasing interest in studying the growth of analytic solutions of (1) in the unit disc D = {z : |z| < 1}. For example, finite order solutions have been studied in [3], [13], [9], [19], [1], [15], [17], [4] as well as solution of finite iterated order in [10], [2]. For r > 0 ∈ D define the iterations exp 1 r = e r , exp n+1 r = exp(exp n r), n ∈ N, and log + = max{log x, 0}, log + 1 r = log + r, log + n+1 r = log + log + n r, n ∈ N. For p ∈ N ∪ {0} the p-th iterated order of an analytic function f in D is defined by…”

Zero distribution of solutions of complex linear differential equations determines growth of coefficients

“…where δ kk = 0 and δ ki = 1 otherwise. [5]. Part (b) is not explicitly stated in [5] and the proof requires some technical manoeuvres.…”

A unit disc analogue of the Bank-Laine conjecture does not hold

“…is the logarithmic order of A(z) [7]. This calculation requires a fair amount of work, but the details are essentially worked out in [7, pp.…”