Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 66, 1-13; http://www.math.u-szeged.hu/ejqtde/

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

Jin Tu^{1} and Teng Long^{1}

1College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China

Abstract

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. Keywords: linear differential equations; entire functions; meromorphic functions; iterated order. AMS Subject Classification (2000): 30D35 , 34A20

1. 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

σ_{p}(f) = \overline{lim_{r→∞}} \frac{log_{p} T (r, f)}{log r} (p ∈ N).  (1.1)

Remark 1.1. If p = 1, the classical growth of order of f is defined by (see [6, 9])

σ(f) = \overline{lim_{r→∞}} \frac{log T (r, f)}{log r} = \overline{lim_{r→∞}} \frac{log_{2} M(r, f)}{log r} (p ∈ N).

∗Corresponding author: E-mail: tujin2008@sina.com.

If p = 2, the hyper-order of f is defined by (see [13])

σ_{2}(f) = \overline{lim_{r→∞}} \frac{log_{2} T (r, f)}{log r} = \overline{lim_{r→∞}} \frac{log_{3} M(r, f)}{log r} (p ∈ N).

If f is an entire function, then the p-iterated order of f is defined by

σ_{p}(f) = \overline{lim_{r→∞}} \frac{log_{p} T (r, f)}{log r} = \overline{lim_{r→∞}} \frac{log_{p+1} M(r, f)}{log r} (p ∈ N).  (1.2)

Definition 1.2. The finiteness degree of the order of a meromorphic function f is defined by

i(f) = \{ \begin{matrix}0 & for f rational, \\ min\{p ∈ N : σ_{p}(f) < ∞\} & for f transcendental for which some \\ & p ∈ N with σ_{p}(f) < ∞ exists , \\ ∞ & for f with σ_{p}(f) = ∞ for all p ∈ N. \end{matrix}  (1.3)

Definition 1.3. The p-iterated lower order of a meromorphic function f is defined by

μ_{p}(f) = lim_{\overline{r→∞}} \frac{log_{p} T (r, f)}{log r} (p ∈ N).  (1.4)

Definition 1.4. The p-iterated exponent of convergence of a-point of a meromorphic function f

is defined by

λ_{p}(f, a) = \overline{lim_{r→∞}} \frac{log_{p} n(r, a)}{log r} = \overline{lim_{r→∞}} \frac{log_{p} N(r, a)}{log r} (p ∈ N).  (1.5)

If a = 0, the p-iterated exponent of convergence of zero-sequence of a meromorphic function f is defined by

λ_{p}(f) = \overline{lim_{r→∞}} \frac{log_{p} n(r, \frac{1}{f})}{log r} = \overline{lim_{r→∞}} \frac{log_{p} N(r, \frac{1}{f})}{log r} (p ∈ N).  (1.6)

The p-iterated exponent of convergence of different zero-sequence of a meromorphic function f is defined by

\overline{λ}_{p}(f) = \overline{lim_{r→∞}} \frac{log_{p} \overline{n}(r, \frac{1}{f})}{log r} = \overline{lim_{r→∞}} \frac{log_{p} \overline{N}(r, \frac{1}{f})}{log r} (p ∈ N).  (1.7)

If a = ∞, the p-iterated exponent of convergence of pole-sequence of a meromorphic function f is defined by

λ_{p} (\frac{1}{f} ) = \overline{lim_{r→∞}} \frac{log_{p} n(r, f)}{log r} = \overline{lim_{r→∞}} \frac{log_{p} N(r, f)}{log r} (p ∈ N).  (1.8)

2. Introduction and Main results

Many authors have investigated complex oscillation properties of the high order linear differential equations

f^{(k)} + A_{k−1}(z)f^{(k−1)} + ⋯ + A_{0}(z)f = 0  (2.1)

and

f^{(k)} + A_{k−1}(z)f^{(k−1)} + ⋯ + A_{0}(z)f = F (z)  (2.2)

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.

Theorem A [10]. Let A_{0}(z), ⋯, A_{k−1}(z) be entire functions, if i(A_{j}) ≤ p(j = 0, ⋯, k−1, p ∈ N), then σ_{p+1}(f) ≤ max{σ_{p}(A_{j}), j = 0, ⋯, k − 1} holds for all solutions of (2.1).

Theorem B [10]. Let A_{0}(z), ⋯, A_{k−1}(z) be entire functions and let i(A_{0}) = p, p ∈ N. If i(A_{j}) < p or σ_{p}(A_{j}) < σ_{p}(A_{0}) for all j = 1, ⋯, k − 1, then i(f) = p + 1 and σ_{p+1}(f) = σ_{p}(A_{0}) hold for all non-trivial solutions of (2.1).

Theorem C [1]. Let A_{0}(z), …, A_{k−1}(z) be entire functions, and let i(A_{0}) = p (p ∈ N). Assume that max{ρ_{p}(A_{j}) : j = 1, 2, …, k − 1} ≤ σ_{p}(A_{0})(> 0) and max{τ_{p}(A_{j}) : σ_{p}(A_{j}) = σ_{p}(A_{0})} < τ_{p}(A_{0}) = τ (0 < τ < +∞). Then every solution f ≢ 0 of (2.1) satisfies i(f) = p + 1 and σ_{p+1}(f) = σ_{p}(A_{0}).

When the coefficients in (2.1) or (2.2) are meromorphic functions of finite iterated order, we have also the following results.

Theorem D [3,11]. Let A_{0}(z), ⋯, A_{k−1}(z), F ≢ 0 be meromorphic functions, and let f(z) be a meromorphic solution of (2.2) satisfying one of the following conditions:

(i) max{i(F) = q, i(A_{j})(j = 0, ⋯, k − 1)} < i(f) = p + 1(p ∈ N) ,

(ii) b = max{σ_{p+1}(F), σ_{p+1}(A_{j})(j = 0, …, k − 1)} < σ_{p+1}(f), then \overline{λ}_{p+1}(f) = λ_{p+1}(f) = σ_{p+1}(f).

Theorem E [11]. Let A_{j}(z)(j = 0, ⋯, k − 1) be meromorphic functions of finite iterated order satisfying β = max\{σ_{p}(A_{j}), λ_{p}(\frac{1}{A_{0}} ), j ≠ 0\} < σ_{p}(A_{0}), (p ∈ N), or i(A_{j}) < p for j ≠ 0, if f ≢ 0 is a meromorphic solution of (2.1), then σ_{p+1}(f) ≥ σ_{p}(A_{0}).

Theorem F [11]. LetA_{j}(z)(j = 0, ⋯, k − 1) be meromorphic functions of finite iterated order satisfying max\{σ_{p}(A_{j}), λ_{p}(\frac{1}{A_{s}}), j ≠ s\} < μ_{p}(A_{s}) ≤ σ_{p}(A_{s}) < ∞, s ∈ \{0, 1, ⋯, k − 1\}(p ∈ N), or

i(A_{j}) < p for j ≠ s, if f ≢ 0 is a meromorphic solution of (2.1) satisfying \frac{N(r,f)}{\overline{N}(r,f)} < M (a positive constant), then σ_{p+1}(f) ≤ σ_{p}(A_{s}).

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.

Theorem 2.1. Let A_{0}(z), ⋯, A_{k−1}(z) be entire functions of finite iterated order satisfying

i(A_{0}) = p, σ_{p}(A_{0}) = σ_{1} and \overline{lim_{r→∞}} ∑_{j=1}^{k−1} m(r, A_{j})/m(r, A_{0}) < 1, then every non-trivial solution f(z)

of (2.1) satisfies σ_{p+1}(f) = σ_{p}(A_{0}) = σ_{1}.

Corollary 2.1. Let A_{0}(z), A_{1}(z), ⋯, A_{k−1}(z) be entire functions of finite iterated order satis-

fying \overline{lim_{r→∞}} ∑_{j=1}^{k−1} m(r, A_{j})/m(r, A_{0}) < 1 and A_{0}(z) be transcendental with σ(A_{0}) < ∞, then every

non-trivial solution f(z) of (2.1) satisfies σ_{2}(f) = σ(A_{0}).

Theorem 2.2. Let A_{0}(z), A_{1}(z), ⋯, A_{k−1}(z) be entire functions of finite iterated order satisfy-

ing max\{σ_{p}(A_{j}), j ≠ 0\} ≤ μ_{p}(A_{0}) = σ_{p}(A_{0}), and lim_{\overline{r→∞}} ∑_{j=1}^{k−1} m(r, A_{j})/m(r, A_{0}) < 1(r ∉ E_{1}), where

E_{1} is a set of r of finite linear measure, then every non-trivial solution f(z) of (2.1) satisfies σ_{p+1}(f) = μ_{p}(A_{0}) = σ_{p}(A_{0}).

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 A_{j}(z)(j = 1, ⋯, k − 1) and obtain the same conclusion σ_{p+1}(f) = σ_{p}(A_{0})(p ∈ N). We have to note that the condition max{σ_{p}(A_{j}), j = 1, ⋯, k − 1} < σ_{p}(A_{0}) in Theorem B

is stronger than our condition \overline{lim_{r→∞}} ∑_{j=1}^{k−1} m(r, A_{j})/m(r, A_{0}) < 1 in Theorem 2.1. Thus, Theo-

rem 2.1 is an improvement of Theorem B and Corollary 2.1 is an improvement of [5, p.121,

Theorem 4]. In Theorem 2.1, if we replace the condition \overline{lim_{r→∞}} ∑_{j=1}^{k−1} m(r, A_{j})/m(r, A_{0}) < 1 with lim_{\overline{r→∞}} ∑_{j=1}^{k−1} 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.

Theorem 2.3. Let A_{0}(z), A_{1}(z), ⋯, A_{k−1}(z), F (z) ≢ 0 be meromorphic functions. If f(z) is a

meromorphic solution of (2.2) satisfying i(f) = p+1, σ_{p+1}(f) = σ_{2} and \overline{lim_{r→∞}}[ ∑_{j=0}^{k−1} T (r, A_{j}) + T (r, F)]

/T (r, f) < 1, then \overline{λ}_{p+1}(f) = λ_{p+1}(f) = σ_{p+1}(f) = σ_{2}.

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.

Theorem 2.4. Let A_{0}(z), A_{1}(z), ⋯, A_{k−1}(z) be meromorphic functions of finite iterated order

satisfying i(A_{0}) = p, δ(∞, A_{0}) = lim_{\overline{r→∞}} \frac{m(r,A_{0})}{T (r,A_{0})} > 0 and \overline{lim_{r→∞}} ∑_{j=1}^{k−1} m(r, A_{j})/m(r, A_{0}) < 1, then every

non-trivial solution f(z) of (2.1) satisfies σ_{p+1}(f) ≥ σ_{p}(A_{0}).

Theorem 2.5. Let A_{0}(z), A_{1}(z), ⋯, A_{k−1}(z), F (z) be meromorphic functions of finite iterated order satisfying max\{σ_{p}(A_{j}), σ_{p}(F), λ_{p}(\frac{1}{A_{s}}), j ≠ s\} < μ_{p}(A_{s}) ≤ σ_{p}(A_{s}) or i(A_{j}) < p (j ≠ s),

if f(z) is a meromorphic solution of (2.2) satisfying \frac{N(r,f)}{\overline{N}(r,f)} ≤ exp_{p−1}\{r^{b}\}(b < μ_{p}(A_{s})), then σ_{p+1}(f) ≤ σ_{p}(A_{s}).

Corollary 2.2. Let A_{0}(z), A_{1}(z), ⋯, A_{k−1}(z) be meromorphic functions of finite iterated order satisfying max\{σ_{p}(A_{j}), λ_{p}(\frac{1}{A_{0}} ), j ≠ 0\} < μ_{p}(A_{0}) ≤ σ_{p}(A_{0}) < ∞, if f ≢ 0 is a meromorphic solu-

tion of (2.1) satisfying \frac{N(r,f)}{\overline{N}(r,f)} < exp_{p−1}\{r^{b}\} (b < μ_{p}(A_{0})), then σ_{p+1}(f) = σ_{p}(A_{0}).

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.

3. Preliminary Lemmas

Lemma 3.1 [9]. Let g : (0, +∞) ⟶ R, h : (0, +∞) ⟶ R be monotone increasing functions such that

(i) g(r) ≤ h(r) outside of an exceptional set E_{2} of finite linear measure. Then, for any α > 1, there exists r_{0} > 0 such that g(r) ≤ h(αr) for all r > r_{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 |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

\frac{f^{(n)}(z)}{f(z)} = (\frac{ν_{f}(r)}{z})^{n} (1 + o(1)) (n ∈ N, r ∉ E_{3}),  (3.1)

where ν_{f}(r) is the central index of f.

Lemma 3.3 [11]. Let f(z) = \frac{g(z)}{d(z)}, 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

\frac{f^{(n)}(z)}{f(z)} = (\frac{ν_{g}(r)}{z})^{n} (1 + o(1)) (n ∈ N).  (3.2)

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

\overline{lim_{r→∞}} \frac{log_{p} ν_{f}(r)}{log r} = σ_{3},  (3.3)

lim_{\overline{r→∞}} \frac{log_{q} ν_{f}(r)}{log r} = μ.  (3.4)

Lemma 3.5 [8]. Let A_{0}(z), A_{1}(z), ⋯, A_{k−1}(z) be entire coefficients in (1.1), and at least one of them is transcendental. If A_{s}(z)(0 ≤ s ≤ k − 1) is the first one (according to the sequence

of A_{0}, A_{1}, ⋯, A_{k−1}) satisfying lim_{\overline{r→∞}} ∑_{j=s+1}^{n} m(r, A_{j})/m(r, A_{s}) < 1(r ∉ E_{5}), where E_{5} ⊂ (1, +∞)

is a set having finite linear measure, then (1.1) possesses at most s linearly independent entire solutions satisfying \overline{lim_{r→∞}} \frac{log T (r,f)}{m(r,A_{p})} = 0(r ∉ E_{5}).

Lemma 3.6 [14]. Let f(z) be an entire function of finite iterated order with i(f) = p, p ∈ N. Then there exist entire functions β(z) and D(z) such that

f(z) = β(z)e^{D(z)},
σ_{p}(f) = max{σ_{p}(β), σ_{p}(e^{D(z)})}

and

σ_{p}(β) = \overline{lim_{r→∞}} \frac{log_{p} N(r, \frac{1}{f})}{log r}.

Moreover, for any given ε > 0, then

log |β(z)| ≥ − exp_{p−1}{r^{σ_{p}(β)+ε}} (r ∉ E_{6}),  (3.5)

where E_{6} is a set of r of finite linear measure.

Lemma 3.7. Let f(z) be a meromorphic function of finite iterated order with i(f) = p, p ∈ N. Then there exist entire functions π_{1}(z), π_{2}(z) and D(z) such that

f(z) = \frac{π_{1}(z)e^{D(z)}}{π_{2}(z)} ,  (3.6)

and

σ_{p}(f) = max{σ_{p}(π_{1}), σ_{p}(π_{2}), σ_{p}(e^{D(z)})}.  (3.7)

Moreover, for any given ε > 0, we have

exp{− exp_{p−1} {r^{σ_{p}(f)+ε}}} ≤ |f(z)| ≤ exp_{p} {r^{σ_{p}(f)+ε}} (r ∉ E_{7}),  (3.8)

where E_{7} is a set of r of finite linear measure. P roof. By Lemma 3.6, it is easy to see that (3.6) and (3.7) hold. Set f(z) = \frac{π_{1}(z)e^{D(z)}}{π_{2}(z)} , where π_{1}(z), π_{2}(z) are the canonical products formed with the zeros and poles of f(z) respectively. Since max{σ_{p}(π_{1}), σ_{p}(π_{2}), σ_{p}(e^{D(z)})} = σ_{p}(f) and by Lemma 3.6, for sufficiently large |z| = r, we have

|π_{1}(z)| ≤ exp_{p} \{r^{σ_{p}(π_{1})+ \frac{ε}{2}} \} ≤ exp_{p} \{r^{σ_{p}(f)+ \frac{ε}{2}} \}, |e^{D(z)}| ≤ exp_{p} \{r^{σ_{p}(f)+ \frac{ε}{2}} \}  (3.9)
|π_{2}(z)| ≤ exp_{p} \{r^{σ_{p}(π_{2})+ \frac{ε}{2}} \} ≤ exp_{p} \{r^{σ_{p}(f)+ \frac{ε}{2}} \},  (3.10)
|π_{1}(z)| ≥ exp\{− exp_{p−1} \{r^{σ_{p}(π_{1})+ \frac{ε}{2}} \}\} ≥ exp\{− exp_{p−1} \{r^{σ_{p}(f)+ \frac{ε}{2}} \}\} (r ∉ E_{7}),  (3.11)
|π_{2}(z)| ≥ exp\{− exp_{p−1} \{r^{σ_{p}(π_{2})+ \frac{ε}{2}} \}\} ≥ exp\{− exp_{p−1} \{r^{σ_{p}(f)+ \frac{ε}{2}} \}\} (r ∉ E_{7}),  (3.12)

where E_{7} is a set of r of finite linear measure. Since σ_{p−1}(D) = σ_{p}(e^{D(z)}) ≤ σ_{p}(f) and |e^{D(z)}| ≥ e^{−|D(z)|}, for sufficiently large |z| = r, we have

|e^{D(z)}| ≥ e^{−|D(z)|} ≥ exp\{− exp_{p−1} \{r^{σ_{p}(f)+ \frac{ε}{2}} \}\}.  (3.13)

By (3.9)-(3.13), we can easily obtain (3.8). Thus, we complete the proof of Lemma 3.7.

Remark 3.1. Lemma 3.7 is an improvement of Lemma 3.6 and extends the conclusion of [4, p.84, Lemma 4].

Lemma 3.8. Let f(z) be a meromorphic function of finite iterated order satisfying i(f) = p, then there exists a set E_{8} ⊂ (1, +∞) having infinite logarithmic measure such that for all r ∈ E_{8}, we have

lim_{r→∞} \frac{log_{p} T (r, f)}{log r} = σ_{p}(f).

P roof. By Definition 1.1, there exists a sequence \{r_{n}\}_{n=1}^{∞} tending to ∞ and satisfying (1+ \frac{1}{n})r_{n} < r_{n+1} such that

lim_{r_{n}→∞} \frac{log_{p} T (r_{n}, f)}{log r_{n}} = σ_{p}(f).  (3.14)

There exists an n_{1} such that for n ≥ n_{1} and for any r ∈ [r_{n}, (1 + \frac{1}{n})r_{n}], we have

\begin{matrix}log & p & T (r_{n}, f) \\ log & (1 + \frac{1}{n})r_{n} \end{matrix} ≤ \frac{log_{p} T (r, f)}{log r} ≤ \frac{log_{p} T ((1 + \frac{1}{n})r_{n}, f)}{log r_{n}}.  (3.15)

Set E_{8} = ⋃_{n=n_{1}}^{∞}[r_{n}, (1 + \frac{1}{n})r_{n}], for any r ∈ E_{8}, by (3.15), we have

lim_{r→∞} \frac{log_{p} T (r, f)}{log r} = lim_{r_{n}→∞} \frac{log_{p} T (r_{n}, f)}{log r_{n}} = σ_{p}(f),

and m_{l}E_{8} = ∑_{n=n_{1}}^{∞} ∫_{r_{n}}^{(1+ \frac{1}{n})r_{n}} \frac{dt}{t} = ∑_{n=n_{1}}^{∞} log(1 + \frac{1}{n}) = ∞.

4. Proofs of Theorems.

Proof of Theorem 2.1. We divide the proof into two parts: (i)σ_{p+1}(f) ≥ σ_{1}, (ii)σ_{p+1}(f) ≤ σ_{1}. (i) By (2.1), we get

−A_{0} = \frac{f^{(k)}(z)}{f(z)} + A_{k−1} \frac{f^{(k−1)}(z)}{f(z)} + ⋯ + A_{1} \frac{f′(z)}{f(z)}.  (4.1)

By the lemma of the logarithmic derivative and (4.1), we have

m(r, A_{0}) ≤ ∑_{j=1}^{k−1} m(r, A_{j}) + O{log(rT (r, f))} (r ∉ E),  (4.2)

where E is a set of r of finite linear measure, not necessarily the same at each occurrence. Suppose that

\overline{lim_{r→∞}}∑_{j=1}^{k−1} m(r, A_{j})/m(r, A_{0}) = α < β_{1} < 1,

then for sufficiently large r, we have

∑_{j=1}^{k−1} m(r, A_{j}) < β_{1}m(r, A_{0}).  (4.3)

By (4.2) and (4.3), we have

(1 − β_{1})m(r, A_{0}) ≤ O{log(rT (r, f))} (r ∉ E).  (4.4)

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

(1 − β_{1}) exp_{p−1}{r^{σ_{1}−ε}} ≤ (1 − β_{1})m(r, A_{0}) ≤ O{log(rT (r, f))}.  (4.5)

By (4.5), we have σ_{p+1}(f) ≥ σ_{p}(A_{0}) = σ_{1}.

(ii) By (2.1), we get

|\frac{f^{(k)}(z)}{f(z)}| ≤ |A_{k−1}| |\frac{f^{(k−1)}(z)}{f(z)}| + ⋯ + |A_{1}| |\frac{f′(z)}{f(z)}| + |A_{0}|.  (4.6)

By Lemma 3.2 and (4.6), for all z satisfying |z| = r ∉ E_{3} and |f(z)| = M(r, f), we have

|(\frac{v_{f}(r)}{z})^{k} (1 + o(1))| ≤ (|A_{k−1}| + |A_{k−2}| + ⋯ + |A_{0}|) |(\frac{v_{f}(r)}{z})^{k−1} (1 + o(1))| (r ∉ E_{3}), (4.7)

where E_{3} is a set of r of finite logarithmic linear measure. By (4.3) and σ_{p}(A_{0}) = σ_{1}, it is easy to see that σ_{p}(A_{j}) ≤ σ_{1}(j = 1, ⋯, k − 1). 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

|(\frac{v_{f}(r)}{z})^{k} (1 + o(1))| ≤ k exp_{p}\{r^{σ_{1}+ε}\} |(\frac{v_{f}(r)}{z})^{k−1} (1 + o(1))| .  (4.8)

By Lemma 3.1, Lemma 3.4 and (4.8), we have σ_{p+1}(f) ≤ σ_{1}.

From (i) and (ii), we have that every non-trivial solution f(z) of (2.1) satisfies σ_{p+1}(f) = σ_{p}(A_{0}) = σ_{1}.

Proof of Theorem 2.2. By Lemma 3.5, we obtain that every linearly independent solution f of (2.1) satisfying \overline{lim_{r→∞}} \frac{log T (r,f)}{m(r,A_{0})} > 0(r ∉ E_{1}). This means that every solution f ≢ 0 of (2.1)

satisfying \overline{lim_{r→∞}} \frac{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

σ_{p+1}(f) ≥ μ_{p}(A_{0}) = σ_{p}(A_{0}).  (4.10)

On the other hand, by Theorem A, we have that every solution f ≢ 0 of (2.1) satisfying

σ_{p+1}(f) ≤ max{σ_{p}(A_{j}), j = 0, ⋯, k − 1} = σ_{p}(A_{0}).  (4.11)

By (4.10) and (4.11), we have σ_{p+1}(f) = μ_{p}(A_{0}) = σ_{p}(A_{0}).

Proof of Theorem 2.3. From (2.2), we get

\frac{1}{f} = \frac{1}{F} (\frac{f^{(k)}(z)}{f(z)} + A_{k−1} \frac{f^{(k−1)}(z)}{f(z)} + ⋯ + A_{1} \frac{f′(z)}{f(z)} + A_{0}) .  (4.12)

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, \frac{1}{f} ) ≤ k\overline{n} (r, \frac{1}{f} ) + n (r, \frac{1}{F} ) + ∑_{j=0}^{k−1} n(r, A_{j}),  (4.13)

and

N (r, \frac{1}{f} ) ≤ k\overline{N} (r, \frac{1}{f} ) + N (r, \frac{1}{F} ) + ∑_{j=0}^{k−1} N(r, A_{j}).  (4.14)

By the lemma of the logarithmic derivative and (4.12), we have

m(r, \frac{1}{f}) ≤ m(r, \frac{1}{F}) + ∑_{j=0}^{k−1} m(r, A_{j}) + O\{log(rT (r, f))\} (r ∉ E).  (4.15)

By (4.14) and (4.15), we get

T (r, f) = T (r, \frac{1}{f})+o(1) ≤ k\overline{N}(r, \frac{1}{f})+T (r, F)+∑_{j=0}^{k−1} T (r, A_{j})+O\{log(rT (r, f))\} (r ∉ E). (4.16)

Suppose that

\overline{lim_{r→∞}} \frac{∑_{j=0}^{k−1} T (r, A_{j}) + T (r, F)}{T (r, f)} = δ < c < 1,  (4.17)

by (4.17), for sufficiently large r and for any given ε(0 < ε < c − δ), we have

∑_{j=0}^{k−1} T (r, A_{j}) + T (r, F) ≤ (δ + ε)T (r, f) < cT (r, f).  (4.18)

Substituting (4.18) into (4.16), we get

T (r, f) ≤ k\overline{N}(r, \frac{1}{f}) + cT (r, f) + εT (r, f) (r ∉ E),  (4.19)

by (4.19), we get

T (r, f) ≤ \frac{k}{1 − c − ε}\overline{N}(r, \frac{1}{f}) ≤ \frac{2k}{1 − c}\overline{N}(r, \frac{1}{f}) (r ∉ E),  (4.20)

by Lemma 3.1 and (4.20), we have \overline{λ}_{p+1}(f) ≥ σ_{p+1}(f) = σ_{2}. Hence

\overline{λ}_{p+1}(f) = λ_{p+1}(f) = σ_{p+1}(f) = σ_{2}.

Proof of Theorem 2.4. Suppose that f = \frac{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

(1 − β_{1})m(r, A_{0}) ≤ O{log(rT (r, f))} (r ∉ E).  (4.21)

Since i(A_{0}) = p and by Lemma 3.8, we have

lim_{r→∞} \frac{log_{p} T (r, A_{0})}{log r} = σ_{p}(A_{0}) (r ∈ E_{8}),  (4.22)

where E_{8} is a set of r of infinite logarithmic linear measure. Since δ(∞, A_{0}) = lim_{\overline{r→∞}} \frac{m(r,A_{0})}{T (r,A_{0})} > 0, we have

lim_{r→∞} \frac{log_{p} m(r, A_{0})}{log r} = σ_{p}(A_{0}) (r ∈ E_{8}).  (4.23)

By (4.21) and (4.23), we get σ_{p+1}(f) ≥ σ_{p}(A_{0}).

Proof of Theorem 2.5. Suppose that f = \frac{g(z)}{d(z)} is a meromorphic solution of (2.2), then by (2.2), we get

−A_{s} = \frac{f^{(k)}}{f^{(s)}} + ⋯ + A_{s+1} \frac{f^{(s+1)}}{f^{(s)}} + A_{s−1} \frac{f^{(s−1)}}{f^{(s)}} + ⋯ + A_{0} \frac{f}{f^{(s)}} − \frac{F}{f^{(s)}} .  (4.24)

By (4.24), we have

T (r, A_{s}) ≤ MT (r, f) + ∑_{j≠s} T (r, A_{j}) + O{log(rT (r, f))}, (r ∉ E)  (4.25)

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

\overline{λ}_{p} (\frac{1}{f} ) = \overline{λ}_{p}(d) ≤ max \{λ_{p} (\frac{1}{A_{j}} ) , λ_{p} (\frac{1}{F} ) , (j = 0, ⋯, k − 1)\} < μ_{p}(A_{s}),  (4.26)

and by \frac{N(r,f)}{\overline{N}(r,f)} < exp_{p−1}\{r^{b}\}, where b < μ_{p}(A_{s}), we have N(r, f) < \overline{N}(r, f) exp_{p−1}\{r^{b}\}. Hence

λ_{p}(d) = λ_{p} (\frac{1}{f} ) ≤ max\{\overline{λ}_{p} (\frac{1}{f} ) , b\} < μ_{p}(A_{s}).  (4.27)

By (4.27) and μ_{p}(f) ≥ μ_{p}(A_{s}), we have μ_{p}(g) = μ_{p}(f) ≥ μ_{p}(A_{s}). By (2.2) again, we have

|\frac{f^{(k)}(z)}{f(z)}| ≤ |A_{k−1}| |\frac{f^{(k−1)}(z)}{f(z)}| + ⋯ + |A_{s}| |\frac{f^{(s)}(z)}{f(z)}| + ⋯ + |A_{0}| + |\frac{F (z)}{f(z)}| .  (4.28)

By Lemma 3.3, there exists a set E_{4} having finite logarithmic measure such that for all z satisfying |z| = r ∉ E_{4} and |g(z)| = M(r, g), we have

\frac{f^{(j)}(z)}{f(z)} = (\frac{ν_{g}(r)}{z})^{j} (1 + o(1)) (j = 1, ⋯, k).  (4.29)

Meanwhile for all z satisfying |z| = r ∉ E_{4} and |g(z)| = M(r, g) > 1, we have

|\frac{F (z)}{f(z)}| = |\frac{F (z) · d(z)}{g(z)}| ≤ M · exp_{p}\{r^{σ_{p}(A_{s})}\}.  (4.30)

By Lemma 3.7,there exists a set E_{7} having finite linear measure such that for all z satisfying |z| = r ∉ E_{7} and for any given ε > 0, we have

|A_{j}(z)| ≤ exp_{p}{r^{σ_{p}(A_{s})+ε}}  (j = 0, ⋯, k − 1).  (4.31)

Substituting (4.29)-(4.31) into (4.28), we get

(\frac{ν_{f}(r)}{z})^{k} (1 + o(1)) ≤ k (\frac{ν_{f}(r)}{z})^{k−1} (1 + o(1)) exp_{p}\{r^{σ_{p}(A_{s})+ε}\}.  (4.32)

Since ε is arbitrary, by (4.32) and Lemma 3.4, we have σ_{p+1}(f) ≤ σ_{p}(A_{s}).

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}).

Acknowledgements

We thank the referee’s useful suggestions that improved our paper. This project is supported by the Youth Foundation of Education Bureau of Jiangxi Province in China (Grant No. GJJ09463), the Natural Science Foundation of Jiangxi Province in China (Grant No. 2008GQS0075) and by the Science and Technology Program of Education Bureau of Jiangxi Province in China (Grant No. GJJ08161).

References

[1] B. Belaidi, Growth and oscillation of solutions to linear differential equations with entire coefficients having the same order, Electron. J. Diff. Eqns. No. 70, (2009), 1-10.

[2] L. G. Bernal, On growth k-order of solutions of a complex homogeneous linear differential equations, Proc. Amer. Math. Soc. 101 (1987), 317-322.

[3] T. B. Cao, Z. X. Chen, X. M. Zheng and J. Tu, On the iterated order of meromorphic solutions of higher order linear differential equations, Ann. Differential Equations 21(2005), No.2, 111-122.

[4] Z. X. Chen, On the hyper order of solutions of some second order linear differential equations, Acta Math. Sinica Ser. B 18 (2002), 79-88.

[5] Z. X. Chen and C. C. Yang, Quantitative estimations on the zeros and growths of entire solutions of linesr differential equations, Complex Variables and Elliptic Equations 42 (2000) 119-133.

[6] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

[7] W. K. Hayman, The local growth of power series: A survey of the Wiman-Valiron method, Canad. Math. Bull. 17 (1974), 317-358.

[8] Y. Z. He and X. Z. Xiao, Algebroid Functions and Ordinary Differential Equations, Science Press, Beijing, 1988 (in Chinese).

[9] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.

[10] L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math. (4) 22 (1998), 385-405.

[11] J. Tu and Z. X, Chen, Growth of solution of complex differential equations with meromorphic coefficients of finite iterated order, Southeast Asian Bull. Math. 33 (2009), 153-164.

[12] J. Tu, Z. X. Chen and X. M. Zheng, Growth of solutions of complex differential equations with coefficients of finite iterated order, Electron. J. Diff. Eqns. No. 54, (2006), 1-8.

[13] C. C. Yang and H. X. Yi, The Uniqueness Theory of Meromorphic Functions, Science Press, Kluwer Academe Publishers, Beijing-New York, 2003.

[14] J. H. Zheng and C. C. Yang, Estimate on the number of fix-points of composite entire functions, Complex Variables and Elliptic Equations 24 (1994), 301-309.

(Received July 18, 2009)