Growth properties of solutions of complex differential equations with entire coefficients of finite (alpha,beta,gamma)-order

. In this article, we investigate the complex higher order linear differential equations in which the coeﬃcients are entire functions of ( α,β,γ )- order and obtain some results which improve and generalize some previous results of Tu et al. [29] as well as Bela¨ıdi [1, 2, 3].


Introduction
Throughout this article, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of entire and meromorphic functions and the theory of complex linear differential equations which are available in [12,21,34] and therefore we do not explain those in details.To study the generalized growth properties of entire and meromorphic functions, the concepts of different growth indicators such as the iterated p-order (see [20,26] ), the (p, q)-th order (see [17,18]), (p, q)-ϕ order (see [27]) etc. are very useful and during the past decades, several authors made close investigations on the generalized growth properties of entire and meromorphic functions related to the above growth indicators in some different directions.The theory of complex linear equations has been developed since 1960s.Many authors have investigated the complex linear differential equations and achieved many valuable results when the coefficients A 0 (z), . . ., A k−1 (z), F (z) (k ≥ 2) in (1.1) or (1.2) are entire functions of finite order or finite iterated p-order or (p, q)-th order or (p, q)-ϕ order; see [1,2,3,7,8,10,15,21,22,23,24,26,27,29,30,31,33].
In [9], Chyzhykov and Semochko showed that both definitions of iterated porder and the (p, q)-th order have the disadvantage that they do not cover arbitrary growth (see [9,Example 1.4]).They used more general scale, called the ϕ-order (see [9]).In recent times, the concept of ϕ-order is used to study the growth of solutions of complex differential equations which extend and improve many previous results (see [4,5,9,19]).
In [25], Mulyava et al. have used the concept of (α, β)-order or generalized order of an entire function in order to investigate the properties of solutions of a heterogeneous differential equation of the second order and obtained several interesting results.For details about (α, β)-order one may see [25,28].
In this paper, we investigate the complex higher order linear differential equations in which the coefficients are entire functions of (α, β, γ)-order and obtain some results which improve and generalize some previous results of Tu et al. [29] as well as Belaïdi [1,2,3].

Definitions and notation
First of all, let L be a class of continuous non-negative on (−∞, +∞) function α such that α(x) = α(x 0 ) ≥ 0 for x ≤ x 0 and α(x) ↑ +∞ as Particularly, when α ∈ L 3 , then one can easily verify that α(mr) ≤ mα(r), m ≥ 2 is an integer.Up to a normalization, subadditivity is implied by concavity.Indeed, if α(r) is concave on [0, +∞) and satisfies α(0) ≥ 0, then for t ∈ [0, 1], so that by choosing t = a a+b or t = b a+b , we obtain As a non-decreasing, subadditive and unbounded function, α(r) satisfies for any R 0 ≥ 0. This yields that α(r) ∼ α(r + R 0 ) as r → +∞.Now we add two conditions on α, β and γ: Throughout this paper, we assume that α, β and γ always satisfy the above two conditions unless otherwise specifically stated.
Heittokangas et al. [16] introduced a new concept of ϕ-order of entire and meromorphic function considering ϕ as subadditive function.For details one may see [16].Extending this notion, recently Belaïdi and Biswas [6] introduce the definition of the (α, β, γ)-order of a meromorphic function in the following way: [2] M (r, f )) β(log γ(r)) .
The linear measure of a set E ⊂ [0, +∞) is defined as t dt, where χ E (t) is the characteristic function of E. The upper and lower densities of and and

Some Lemmas
In this section we present some lemmas which will be needed in the sequel.
Lemma 4.1 ([11]).Let f (z) be a nontrivial entire function, and let κ > 1 and ε > 0 be given constants.Then there exist a constant c > 0 and a set E 1 ⊂ [0, +∞) having finite linear measure such that for all z satisfying |z| = r / ∈ E 1 , we have 13,14,21,32]).Let f (z) be a transcendental entire function, and let z be a point with |z| = r at which |f (z)| = M (r, f ).Then, for all |z| outside a set E 2 of r of finite logarithmic measure, we have where ν(r, f ) is the central index of f (z).
Thus, from the above we obtain On the other hand, for an entire function f (z), we have f (z) − f (0) = z 0 f (t)dt, where the integral being taken along the straight line from 0 to z, so we obtain that Therefore from above we have log [3] M (r, f ) ≤ log [3] M (r, f ) + log [3] r + log [3] |f (0)| + O(1).
Remark 4.5.In the line of Lemma 4.4 one can easily deduce that σ (α,β,γ , where f (z) is an entire transcendental function.