Iterated Order of Solutions of Linear Differential Equations with Entire Coefficients

In this paper, we investigate the iterated order of solutions of higher order homogeneous linear differential equations with entire coef- ficients. We improve and extend some results of Bela¨odi and Hamouda by using the concept of the iterated order. We also consider nonhomogeneous linear differential equations.


Introduction and main results
In this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions (see [13]).In addition, we use the notations σ(f ) to denote the order of growth of a meromorphic function f (z).
We define the linear measure of a set E ⊂ [0, +∞) by m(E) = For the definition of the iterated order of a meromorphic function, we use the same definition as in [14] , [5, p. 317] , [15, p. 129] .For all r ∈ R, we define exp 1 r := e r and exp p+1 r := exp exp p r , p ∈ N. We also define for all r EJQTDE, 2010 No. 32, p. 1 sufficiently large log 1 r := log r and log p+1 r := log log p r , p ∈ N.Moreover, we denote by exp 0 r := r, log 0 r := r, log −1 r := exp 1 r and exp −1 r := log 1 r.Definition 1.1 Let p ≥ 1 be an integer.Then the iterated p−order σ p (f ) of a meromorphic function f (z) is defined by where T (r, f ) is the characteristic function of Nevanlinna.For p = 1, this notation is called order and for p = 2, hyper-order.
Remark 1.1 The iterated p−order σ p (f ) of an entire function f (z) is defined by where M (r, f ) = max The finiteness degree of the order of a meromorphic function f is defined by 0, if f is rational, min {j ∈ N : σ j (f ) < ∞} , if f is transcendental with σ j (f ) < ∞ for some j ∈ N, ∞, if σ j (f ) = ∞ for all j ∈ N.
(1.3) Definition 1. 3 The iterated convergence exponent of the sequence of zeros of a meromorphic function f (z) is defined by where N r, 1 f is the counting function of zeros of f (z) in {z : |z| < r}.Similarly, the iterated convergence exponent of the sequence of distinct zeros of f (z) is defined by EJQTDE, 2010 No. 32, p. 2 where N r, 1 f is the counting function of distinct zeros of f (z) in {z : |z| < r}.
Definition 1.4 The finiteness degree of the iterated convergence exponent of the sequence of zeros of a meromorphic function f (z) is defined by (1.6) Remark 1.2 Similarly, we can define the finiteness degree i λ (f ) of λ p (f ).
Let n ≥ 2 be an integer and let A 0 (z) , ..., A n−1 (z) with A 0 (z) ≡ 0 be entire functions.It is well-known that if some of the coefficients of the linear differential equation are transcendental, then the equation (1.7) has at least one solution of infinite order.Thus, the question which arises is : What conditions on A 0 (z) , ..., A n−1 (z) will guarantee that every solution f ≡ 0 of (1.7) has an infinite order?For the above question, there are many results for the second and higher order linear differential equations (see for example [2] , [3] , [4] , [8] , [11] , [14] , [15]).In 2001 and 2002, Belaïdi and Hamouda have considered the equation (1.7) and have obtained the following two results: Theorem A [4] Let A 0 (z) , ..., A n−1 (z) with A 0 (z) ≡ 0 be entire functions such that for real constants α, β, µ, θ 1 and θ 2 satisfying 0 ≤ β < α, µ > 0 and θ 1 < θ 2 , we have and Then every solution f ≡ 0 of the equation (1.7) has an infinite order.
Let n ≥ 2 be an integer and consider the linear differential equation It is well-known that if A n ≡ 1, then all solutions of this equation are entire functions but when A n is a nonconstant entire function, equation (1.12) can possess meromorphic solutions.For instance the equation . Recently, L. Z. Yang [18], J. Xu and Z. Zhang [17] have considered equation (1.12) and obtained different results concerning the growth of its solutions, but the condition that the poles of every meromorphic solution of (1.12) must be of uniformly bounded multiplicity was missing in [17].See Remark 3 in [9] .
In the present paper, we improve and extend Theorem A and Theorem B for equations of the form (1.12) by using the concept of the iterated order.We also consider the nonhomogeneous linear differential equations.We obtain the following results: Theorem 1.1 Let p ≥ 1 be an integer and let A 0 (z) , ..., A n−1 (z) , A n (z) with A 0 (z) ≡ 0 and A n (z) ≡ 0 be entire functions such that i λ (A n ) ≤ 1, i (A j ) = EJQTDE, 2010 No. 32, p. 4 p (j = 0, 1, ..., n) and max {σ p (A j ) : j = 1, 2, ..., n} < σ p (A 0 ) = σ.Suppose that for real constants α, β, θ 1 and θ 2 satisfying 0 ≤ β < α and θ 1 < θ 2 and for ε > 0 sufficiently small, we have and Then every meromorphic solution f ≡ 0 whose poles are of uniformly bounded multiplicity of the equation (1.12) has an infinite iterated p−order and satisfies i (f and two real numbers α and β satisfying 0 ≤ β < α such that for ε > 0 sufficiently small, we have and as k → +∞.Then every meromorphic solution f ≡ 0 whose poles are of uniformly bounded multiplicity of the equation (1.12) has an infinite iterated p−order and satisfies i (f z) be entire functions with A 0 (z) ≡ 0, A n (z) ≡ 0 and F ≡ 0. Considering the nonhomogeneous linear differential equation we obtain the following result: EJQTDE, 2010 No. 32, p. 5 Theorem 1.3 Let A 0 (z) , ..., A n−1 (z) ,A n (z) with A 0 (z) ≡ 0 and A n (z) ≡ 0 be entire functions satisfying the hypotheses of Theorem 1.2 and let F ≡ 0 be an entire function of iterated order with i (F ) = q.

Preliminary Lemmas
Lemma 2.1 [10] Let f (z) be a meromorphic function.Let α > 1 and Then there exist a set E 1 ⊂ (1, +∞) having finite logarithmic measure and a constant B > 0 that depends only on α and Γ such that for all z satisfying |z| = r / ∈ [0, 1] ∪ E 1 and all (k, j) ∈ Γ, we have ) and for sufficiently large k, we have and where ν g (r) is the central index of g.

Proof of Theorem 1.1
Suppose that f ( ≡ 0) is a meromorphic solution whose poles are of uniformly bounded multiplicity of the equation (1.12).From (1.12), it follows that By Lemma 2.2, there exist a constant B > 0 and a set E 2 ⊂ [0, +∞) having finite linear measure such that for all z satisfying |z| = r / ∈ E 2 , we have 3), we obtain that σ p (f ) = +∞ and i (f We can rewrite (1.12) as Obviously, the poles of f (z) can only occur at the zeros of A n (z).Note that the multiplicity of the poles of f is uniformly bounded, and thus we have i λ Thus by Lemma 2.4, there exists a sequence of complex numbers {z k } k∈N and a set E 3 of finite logarithmic measure such that ) and for sufficiently large k, we have By Remark 1.1, for any given ε > 0 and for sufficiently large r, we have By Lemma 2.5, for the above ε > 0, there exists a set E 4 ⊂ [1, +∞) that has finite linear measure and finite logarithmic measure such that for all z We can rewrite (1.12) as Substituting (3.4) into (3.7),we obtain for the above z k Hence from (3.5), (3.6) and (3.8), we have where . By Lemma 2.9 and (3.9), we get lim sup Since ε > 0 is arbitrary, by (3.10) and Lemma 2.3, we obtain i (f ) = i (g) ≤ p + 1 and σ p+1 (f ) = σ p+1 (g) ≤ σ.This and the fact that 4 Proof of Theorem 1.2 Suppose that f ( ≡ 0) is a meromorphic solution whose poles are of uniformly bounded multiplicity of the equation (1.12).From (1.12), it follows that By Lemma 2.2, there exist a constant B > 0 and a set E 2 ⊂ [0, +∞) having finite linear measure such that for all z satisfying |z| = r / ∈ E 2 , we have Hence from (1.15), (1.16), (4.1) and (4.2), we have Hence from (4.3) and Lemma 2.8, we obtain that σ p (f ) = +∞ and i (f ) ≥ p+1, σ p+1 (f ) ≥ σ −ε.Since ε > 0 is arbitrary, we get σ p+1 (f ) ≥ σ.By using the same arguments as in proof of Theorem 1.1, we obtain i (f ) ≤ p + 1 and σ p+1 (f ) ≤ σ.Hence i (f ) = p + 1 and σ p+1 (f ) = σ.EJQTDE, 2010 No. 32, p. 12 First, we show that (1.17) can possess at most one exceptional meromorphic solution f 0 satisfying i (f 0 ) < p + 1 or σ p+1 (f 0 ) < σ.In fact, if f * is another solution with i (f * ) < p + 1 or σ p+1 (f * ) < σ of the equation (1.17), then i (f 0 − f * ) < p + 1 or σ p+1 (f 0 − f * ) < σ.But f 0 − f * is a solution of the corresponding homogeneous equation (1.12) of (1.17).This contradicts Theorem 1.2.We assume that f is an infinite iterated p−order meromorphic solution whose poles are of uniformly bounded multiplicity of (1.17) and f 1 , f 2 , ...f n is a solution base of the corresponding homogeneous equation (1.12) of (1.17).Then f can be expressed in the form where B 1 (z) , ..., B n (z) are suitable meromorphic functions determined by Since the Wronskian W (f 1 , f 2 , ..., f n ) is a differential polynomial in f 1 , f 2 , ..., f n with constant coefficients, it is easy by using Theorem 1.2 to deduce that σ p+1 (W ) ≤ max {σ p+1 (f j ) : j = 1, 2, ..., n} = σ p (A 0 ) = σ.

+∞ 0 χ 1 χ
E (t)dt and the logarithmic measure of a set H ⊂ [1, +∞) by lm(H) = +∞ H (t) t dt, where χ F denote the characteristic function of a set F .