Growth of solutions of second order complex linear differential equations with entire coefficients

Abstract

Some new conditions on the entire coefficients A(z) and B(z), which guarantee every nontrivial solution of f + A(z) f + B(z) f = 0 is of infinite order, are given in this paper. Two classes of entire functions are involved in these conditions, the one is entire functions having Fabry gaps, the another is function extremal for Yang's inequality. Moreover, a kind of entire function having finite Borel exception value is considered.

Order of growth of solutions of linear complex differential equations when the coefficients have same order

“…There are many research articles in which authors used coefficient having a Fabry gaps. Long [13], Kumar and Saini [11] also used Fabrys gap in their articles.…”

“…Combining Theorem 1 from [5] and Lemma 2.2 from [10], we get the following lemma. It is proved by Long [13]. It is the property of finite order entire functions having Fabry gaps.…”

On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

“…The growth of solutions of (1) is very interesting topic after Wittich's work [16], the main tool is Nevanlinna theory of meromorphic functions which can be found in [6,10,18]. Many results have been obtained by many different researchers, for the case of complex plane C, see, for example, [10][11][12][13]17] and therein references, for the case of unit disc D, see, for example [1,2,4,7,14] and therein references. Recently, Fettouch and Hamouda investigated the growth of solutions of equation (1) by using a new idea, in which the coefficients are analytic function except a finite singular point, more details can be found in [3,5].…”

On a question of Gundersen concerning the growth of solutions of linear differential equations

“…The following lemma, originally due to Hille [13,Chapter 7.4], see also [7,20,22], plays an important role in proving Theorem 1.5. The method used in proving the lemma is typically referred to as the method of asymptotic integration.…”