THE ϕ-ORDER OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS IN THE UNIT DISC*

In this paper, we investigate the ( ) pq , -order of solution of second-order linear differential equa- tion ( ) ( ) ( ) '' ' f A zf A z f F z 10 + + = , where ( ) Az 0 , ( ) Az 1 and ( ) Fz are analytic functions in the unit disc. We obtain several theorems about the growth and oscillation of solutions of differential equations.


Introduction and Main results
A classical result due to H. Wittich [13] states that the coefficients of the linear differential equation (1.1) are polynomials if and only if all solutions of (1.1) are entire functions of finite order of growth.Later on, more detailed studies on the growth of solutions were done by different authors; see, for instance, [4,8,11].
In particular, Gundersen, Steinbart and Wang listed all possible orders of growth of entire solutions of (1.1) in terms of the degrees of the polynomial coefficients [5].
Recently, there has been increasing interest in studying the interaction between the analytic coefficients and solutions of (1.1) in the unit disc.The result of Wittich stated above has a natural analogue in the unit disc, as shown in [1,6].For instance, Heittokangas showed that all solutions of (1.1) are finite-order analytic functions in the unit disc if and only if the coefficients are H-functions [6].
A function f, analytic in the unit disc D := {z : |z| < 1}, is an H-function if there exists a q ∈ [0, ∞) such that The space A −∞ , introduced by B. Korenblum [9], coincides with the space of all H-functions.The T -order of a meromorphic function f in D, is defined by where T (r, f ) denotes the Nevanlinna characteristic of f.Equation (1.1) with coefficients in the weighted Bergman spaces are studied in [7,10,12].For 0 < p < ∞ and −1 < α < ∞, the weighted Bergman space B p α consists of those functions f, analytic in D, such that where dm(z) = rdrdθ is the usual Euclidean area measure.Moreover, The following result combines Theorems 1 and 2 in [10]. ( Then all non-trivial solutions f of (1.1) satisfy is the smallest index for which α q = max 0≤j≤k−1 {α j }, then in every fundamental solution base there are at least k − q linearly independent solutions f of (1.1) such that σ T (f ) = α q .
If f is meromorphic in D, then the ϕ-order of f is defined as σ ϕ (f ) = σ ϕ (T (r, f )).The logarithmic order of f is defined as .
Remark 1.1.The usual order of growth of a meromorphic function (ψ) denote the order and the logarithmic order of ψ, respectively.
The following theorem corresponds to Theorem 1.1.
The purpose of this paper is to refine Theorem 1.3.We obtain a result analogous to Theorem 1.2 in terms of the general ϕ-order.In fact, we obtain the following theorem.
Remark 1.2.The case s = 0 of Theorem 1.4 clearly reduces to Theorem 1.3 (2), and the assertion of Theorem 1.4 for s = q is contained in Theorem 1.3 (3).
In order to state the following corollaries of Theorem 1.4, we denote where α k := −1.Moreover, we define Then each solution base of (1.1) admits at most s ⋆ ≤ q solutions f satisfying σ ϕ (f ) < β(s ⋆ ).In particular, there are at most s ⋆ ≤ q solutions f satisfying σ ϕ (f ) = 0.
To estimate the quantity k j=1 σ ϕ (f j ) by using Theorem 1.4, we set γ(j) Note that the sum in (1.4) is considered to be empty, if s ⋆ = q.Corollary 1.2 is sharp.This is illustrated by an example in Section 5.

Lemmas for the proof of Theorem
The following lemma on the order reduction procedure originates from C.
where Let k and j be integers satisfying k > j ≥ 0, and let 0 < δ < 1 and ε > 0. Let f be a meromorphic function in D such that f (j) does not vanish identically.
Here |f (z)| |g(z)| if there exists a constant C > 0 independent of z such that |f (z)| ≤ C|g(z)|.Raising both sides to the power 1 k−1 and integrating θ from 0 to 2π, we obtain,

Example
The sharpness of Corollary 1.2 in the case ϕ(r) = 1  1−r is discussed as follows.For β ≥ 1, the functions f 1,2 (z) = exp{±i( It follows that for the solution base {f 1 , f 2 , f 3 } equality holds in the first inequality in (1.4), and for the solution base {f 1 , f 2 , f 1 + f 3 } equality holds in the last inequality in (1.4).This shows the sharpness of Corollary 1.2 in the case ϕ(r) = 1 1−r .

Acknowledgement
The author would like to thank the referees and editor for their thoughtful comments and helpful suggestions which greatly improved the final version of this paper.