Linear differential equations with analytic coefficients of [p,q]-order in the unit disc

Abstract

In this paper, we investigate the complex higher order linear differential equations in which the coefficients are analytic functions in the unit disc of [p, q]-order. We obtain several theorems about the growth and oscillation of solutions of differential equations.

Second order complex differential equations with analytic coefficients in the unit disc

“…After that, many articles (see e.g. [1,2,3,10,15,16,21]) focused on this topic. In this article, we continue to focus on the same topic by considering the second order complex differential equation…”

Nonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc

“…By using similar proof of Lemma 2.6 in [3] or Lemma 2.6 in [19], we easily obtain the following lemma.…”

“…In 2010, Liu et al [22] firstly studied the growth of solutions of equation (1.1) with entire coefficients of [p, q]-order in the complex plane. After that, many authors applied the concepts of entire (meromorphic) functions in the complex plane and analytic functions in the unit disc ∆ = {z ∈ C : |z| < 1} of [p, q]-order to investigate complex differential equations (see [2]- [5], [19], [21], [23], [24], [26]). In this paper, we use the concept of [p, q]-order to study the growth and zeros of differential polynomial (1.2) generated by meromorphic solutions of [p, q]-order in the unit disc to equation (1.1).…”

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

“…The latter results were generalized on so called [p, q]-orders (see e. g. [19], [1], [15], [17]). But definition p-th iterated order as well as [p, q]-order has the disadvantage that it does not cover arbitrary growth, i. e. there exist functions of infinite p-th iterated order for arbitrary p ∈ N. In the complex plane this case is described in Example 1 in [6].…”

“…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…”