Oscillation of Complex High Order Linear Differential Equations with Coefficients of Finite Iterated Order ∗

In this paper, we investigate the growth of solutions of complex high order linear differential equations with entire or meromorphic coefficients of finite iterated order and we obtain some results which improve and extend some previous results of Z. X. Chen and L. Kinnunen.


Definitions and Notations
In this paper, we assume that readers are familiar with the fundamental results and standard notations of the Nevanlinna's theory of meromorphic functions (see [6,9]).In order to describe the growth of order of entire functions or meromorphic functions more precisely, we first introduce some notations about finite iterated order.Let us define inductively, for r ∈ [0, ∞), exp 1 r = e r and exp i+1 r = exp(exp i r), i ∈ N.For all sufficiently large r, we define log 1 r = log r and log i+1 r = log(log i r), i ∈ N. We also denote exp 0 r = r = log 0 r and exp −1 r = log 1 r.Moreover we denote the linear measure and the logarithmic measure of a set E ⊂ [1, +∞) by mE = E dt and m l E = E dt/t respectively.In the following, we recall some definitions of entire functions or meromorphic functions of finite iterated order (see [2,3,10,12]).
Definition 1.1.The p-iterated order of a meromorphic function f is defined by Remark 1.1.If p = 1, the classical growth of order of f is defined by (see [6,9]) If p = 2, the hyper-order of f is defined by (see [13]) If f is an entire function, then the p-iterated order of f is defined by Definition 1.2.The finiteness degree of the order of a meromorphic function f is defined by min{p ∈ N : σ p (f ) < ∞} for f transcendental for which some If a = 0, the p-iterated exponent of convergence of zero-sequence of a meromorphic function f is defined by The p-iterated exponent of convergence of different zero-sequence of a meromorphic function f is defined by If a = ∞, the p-iterated exponent of convergence of pole-sequence of a meromorphic function f is defined by (1.8)

Introduction and Main results
Many authors have investigated complex oscillation properties of the high order linear differential equations and and obtained many results when the coefficients in (2.1) or (2.2) are entire functions or meromorphic functions of finite order (see [1,2,3,10,11,12]).When the coefficients in (2.1) or (2.2) are entire functions of finite iterated order, we have the following results.
holds for all solutions of (2.1).
When the coefficients in (2.1) or (2.2) are meromorphic functions of finite iterated order, we have also the following results.
Theorem F [11].LetA j (z)(j = 0, • • • , k − 1) be meromorphic functions of finite iterated order satisfying max{σ p (A j ), In this paper, we investigate the growth of solutions of high order linear differential equations (2.1) and (2.2) with entire or meromorphic coefficients of finite iterated order under certain conditions and obtain the following results which improve and extend the above results.
, where E 1 is a set of r of finite linear measure, then every non-trivial solution f (z) of (2.1) satisfies Remark 2.1.In Theorems B-C and our Theorem 2.1, the authors investigated the growth of the solutions of (2.1) under the same case that the coefficient A 0 (z) in (2.1) grows faster than other coefficients m(r, A j )/m(r, A 0 ) < 1 and µ p (A 0 ) = σ p (A 0 ), and then we can get the same conclusion as Theorem 2.1, therefore Theorem 2.2 is a supplement of Theorem 2.1.
Remark 2.2.Theorem 2.3 is an improvement of Theorem D since the conditions in Theorem D are stronger than our condition in Theorem 2.3.Can we get the same conclusion when the coefficients in (2.1) are meromorphic functions?The following Theorem 2.4 give us an affirmative answer.
Remark 2.3.Theorem 2.4 is a supplement of Theorem E. Theorem 2.5 is an extension of Theorem F since Theorem F is a special case of Theorem 2.5 with F (z) ≡ 0.
(ii) g(r) ≤ h(r) outside of an exceptional set E 2 of finite logarithmic measure.Then, for any α > 1, there exists r 0 > 0 such that g(r) ≤ h(r α ) for all r > r 0 .Lemma 3.2 [7,8,9].Let f (z) be a transcendental entire function, and let z be a point with EJQTDE, 2009 No. 66, p. 5 |z| = r at which |f (z)| = M (r, f ).Then for all |z| outside a set E 3 of r of finite logarithmic measure, we have where ν f (r) is the central index of f .
, where g(z), d(z) are entire functions of finite iterated order satisfying i(d) < p or σ p (d) < µ p (g) ≤ σ p (g) < ∞, p ∈ N .Let z be a point with |z| = r at which |g(z)| = M (r, g) and ν g (r) denotes the central-index of g, then the estimation holds for all |z| = r outside a set E 4 of r of finite logarithmic measure.
Lemma 3.4 [3,11].Let f (z) be an entire function of finite iterated order satisfying σ p (f ) = σ 3 , µ q (f ) = µ, 0 < q ≤ p < ∞, and let ν f (r) be the central index of f , then we have where E 6 is a set of r of finite linear measure.
Proof of Theorem 2.1.We divide the proof into two parts: (i) By (2.1), we get By the lemma of the logarithmic derivative and (4.1), we have where E is a set of r of finite linear measure, not necessarily the same at each occurrence.Suppose that then for sufficiently large r, we have By (4.2) and (4.3), we have By σ p (A 0 ) = σ 1 and Lemma 3.8, there exists a set E 8 ⊂ (1, +∞) having infinite logarithmic measure such that for all z satisfying |z| = r ∈ E 8 \E and for any ε(> 0), we have (ii) By (2.1), we get By Lemma 3.2 and (4.6), for all z satisfying |z| = r ∈ E 3 and |f (z)| = M (r, f ), we have where E 3 is a set of r of finite logarithmic linear measure.By (4.3) and . By Lemma 3.7 and (4.7), there exists a set E 7 ⊂ (1, +∞) having finite logarithmic measure such that for all z satisfying |z| = r ∈ (E 3 E 7 ), we have From (i) and (ii), we have that every non-trivial solution f (z) of (2.1) satisfies Proof of Theorem 2.2.By Lemma 3.5, we obtain that every linearly independent solution f of (2.1) satisfying lim r→∞ log T (r,f ) m(r,A 0 ) > 0(r ∈ E 1 ).This means that every solution f ≡ 0 of (2.1) satisfying lim r→∞ log T (r,f ) m(r,A 0 ) > 0(r ∈ E 1 ), then there exist δ > 0 and a sequence {r n } ∞ n=1 tending to ∞ such that for sufficiently large r n ∈ E 1 and for every solution f ≡ 0 of (2.1), we have log T (r n , f ) > δm(r n , A 0 ).(4.9) Since µ p (A 0 ) = σ p (A 0 ) and by (4.9), we have On the other hand, by Theorem A, we have that every solution f ≡ 0 of (2.1) satisfying Proof of Theorem 2.3.From (2.2), we get EJQTDE, 2009 No. 66, p. 9 It is easy to see that if f has a zero at z 0 of order α(> k), and A 0 , • • • , A k−1 are analytic at z 0 , then F must have a zero at z 0 of order α − k, hence n r, and N r, By the lemma of the logarithmic derivative and (4.12), we have .17) by (4.17), for sufficiently large r and for any given ε(0 < ε < c − δ), we have by (4.19), we get by Lemma 3.1 and (4.20), we have EJQTDE, 2009 No. 66, p. 10 Proof of Theorem 2.4.Suppose that f = g(z) d(z) is a non-trivial meromorphic solution of (2.1), by (4.1)-(4.4) in the proof of Theorem 2.1, there exists a constant β 1 < 1 such that for sufficiently large r, we have Proof of Theorem 2.5.Suppose that f = g(z) d(z) is a meromorphic solution of (2.2), then by (2.2), we get By (4.24), we have where M > 0 is a constant, not necessarily the same at each occurrence.By µ p (A s ) > max{σ p (A j )(j = s), σ p (F )} and (4.25), we get µ p (f ) ≥ µ p (A s ).Since the poles of f must be the poles of A j (j = 0, • • • , k − 1) and F , we have  Proof of Corollary 2.2.By the proof of Theorem 2.5, we obtain that every meromorphic solution f ≡ 0 of (2.1) satisfies σ p+1 (f ) ≤ σ p (A 0 ).On the other hand, by Theorem E, we get σ p+1 (f ) ≥ σ p (A 0 ), hence every meromorphic solution f ≡ 0 of (2.1) satisfies σ p+1 (f ) = σ p (A 0 ).