Completely regular growth solutions of second order complex linear differential equations

Abstract

Abstract. If A(z) and B(z) are transcendental entire functions, then all solutions of the differential equation f ′′ + A(z)f ′ + B(z)f = 0 are entire and typically of infinite order. Simple examples show that finite order solutions are also possible. Assuming that A(z) and B(z) are of completely regular growth, Gol'dberg-Ostrovskii-Petrenko asked whether all solutions of finite order are of completely regular growth also. This problem remains unsolved, but several aspects of the problem are addressed. Exponential polynomials form an important subclass of functions of completely regular growth, and they are known to always satisfy certain linear differential equations. Hence cases where the coefficients and/or the solutions are exponential polynomials are also discussed.

Dual exponential polynomials and linear differential equations

“…The following is our main result on the second order equation (4.1), extending [7,Theorem 3.4]. Theorem 1.…”

“…as r → ∞ uniformly in θ. For example, a transcendental exponential polynomial function is of completely regular growth, see [7,Lemma 1.3].…”

“…It is natural to consider linear differential equations with transcendental exponential polynomial coefficients as an important special case of this problem. For the second order case, partial results are given in [7], where exponential polynomial solutions of second order equations with exponential polynomial coefficients are studied.…”

“…In Section 6, we observe some general growth properties of finite order solutions of a class of equations of the form (1.1). In Section 7, we give the proof of our theorem for second order equations, part of which follows from results in [7]. Then in Section 8, the proof of the general theorem for arbitrary order equations is given.…”

Research Questions on Meromorphic Functions and Complex Differential Equations

“…[33, p. 300],[36] Let f be a transcendental solution of (5.1), where A(z) and B(z) have completely regular growth, such that ρ(f ) < ∞. Does f have completely regular growth?…”