On [p,q]-order of growth of solutions of linear di ﬀ erential equations in the unit disc

: The [ p , q ]-order of growth of solutions of the following linear di ﬀ erential equations ( ∗∗ ) is investigated, where A i ( z ) are analytic functions in the unit disc, i = 0 , 1 , ..., k − 1. Some estimations of [ p , q ]-order of growth of solutions of the equation ( ∗∗ ) are obtained when A j ( z ) dominate the others coe ﬃ cients near a point on the boundary of the unit disc, which is generalization of previous results from S. Hamouda.


Introduction and main results
For the following complex linear differential equation where A i (z) are analytic in the unit disc ∆ = {z : |z| < 1}, i = 0, 1, ..., k − 1, k ≥ 2. For the properties of solutions of Eq (1.1), including growth, zero distribution and function space properties, many results have been obtained, for example [9,10,14,16,20,22,23,25] and therein references. In addition, the differential equations have wide applications in various science discipline, for example engineering, predator-prey equations, population growth and decay, Newton's law of cooling and so on, see [1,5,24,28,29] and therein references. This paper is mainly concerned with the growth of solution of the Eq (1.1). It has been widely noted that Bernal [4] firstly introduced the idea of iterated order to express the fast growth of solutions of (1.1). Since then, the iterated order of solutions of (1.1) is very interesting topic in the unit disc ∆, many results concerning iterated order of solutions of (1.1) in the unit disc are obtained, see [3,6,7,18,26]. In order to better estimate the fast growth of solutions of (1.1), [p, q]-order was introduced, and many results on [p, q]-order of solutions of (1.1) have been found by different researchers in ∆. For example, see [2,17,19,21,27]. To state our results, firstly, we assume that readers are familiar with the fundamental results and the standard notation of the Nevanlinna value distribution theory in the unit disc (see [12,13]). Secondly, we introduce some definitions, for all r ∈ [0, 1), exp 1 r = e r and exp n+1 r = exp(exp n r), n ∈ N, and for all r ∈ (0, 1), log 1 r = log r and log n+1 r = log(log n r), n ∈ N. We also denote exp 0 r = r = log 0 r, exp −1 r = log 1 r.
Definition 1. Let f be a meromorphic function in ∆, then the iterated n-order of f is defined by , If f is an analytic in ∆, then its iterated n-order is defined by Suppose p and q are integers satisfying p ≥ q ≥ 1. Then [p, q]-order is defined as follows.
Definition 2. Let f be a meromorphic function in ∆, then the [p, q]-order of f is defined by .
If f is analytic in ∆, then its [p, q]-order is defined by .

Remark 1. [2]
Let p and q be integers such that p ≥ q ≥ 1, and f be an analytic function in ∆. The following two statements holds: Recently, Hamouda [11] considered the fast growing solutions of Eq (1.1) by using a new idea which A 0 dominates the other coefficients near a point on the boundary of the unit disc, and obtained some results which improve and generalize results of Heittokangas et al. [15]. The following two results are proved. Theorem 1.1. [11] Let A 0 (z), . . . , A k−1 (z) be analytic functions in ∆. If there exists ω 0 ∈ ∂∆ and a curve γ ⊂ ∆ tending to ω 0 such that for any constant µ > 0, then every nontrivial solution f (z) of (1.1) is of infinite order.
In Theorems 1.3 and 1.4, the coefficient A 0 (z) is the dominant coefficient. A nature question: How to character [p, q]-order of growth of solutions of Eq (1.1) when A s (z) dominates the other coefficients near a point on the boundary of the unit disc. Next, we study the growth of the solution of (1.1) when A s (z) is the dominant coefficient, and get the following results. Theorem 1.5. Let A 0 (z), . . . , A k−1 (z) be analytic functions in ∆. If there exists ω 0 ∈ ∂∆ and a curve γ ⊂ ∆ tending to ω 0 such that for any µ > 0, we have then every nontrivial solution f (z) of (1.1), in which f (n) (z) just has finite many zeros for all n < s(n = 0, ..., s − 1), is of infinite order.

Conclusions
Many results on [p, q]-order of solutions of (1.1) have been found by different researchers in ∆, in this paper the difference is that we discussed the [p, q]-order of growth of solutions of linear differential Eq (1.1) which A j (z) dominate the others coefficients near a point on the boundary of the unit disc.