Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 32, 1-17; http://www.math.u-szeged.hu/ejqtde/

Iterated Order of Solutions of Linear Differential Equations with Entire Coefficients

Karima HAMANI and Benharrat BELAÏDI

Department of Mathematics Laboratory of Pure and Applied Mathematics University of Mostaganem, B. P. 227 Mostaganem, Algeria hamanikarima@yahoo.fr belaidi@univ-mosta.dz

Abstract. In this paper, we investigate the iterated order of solutions of higher order homogeneous linear differential equations with entire coefficients. We improve and extend some results of Belaïdi and Hamouda by using the concept of the iterated order. We also consider nonhomogeneous linear differential equations.

2010 Mathematics Subject Classification: 34M10, 34M05, 30D35. Key words: Differential equations, Meromorphic function, Iterated order.

1 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) = ∫_{0}^{+∞} χ_{E}(t)dt and the logarithmic measure of a set H ⊂ [1, +∞) by lm(H) = ∫_{1}^{+∞} \frac{χ_{H}(t)}{t}dt, where χ_{F} denote the characteristic function of a set F.

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 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

σ_{p}(f) = lim sup_{r→+∞} \frac{log_{p} T (r, f)}{log r},  (1.1)

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

σ_{p}(f) = lim sup_{r→+∞} \frac{log_{p} T (r, f)}{log r} = lim sup_{r→+∞} \frac{log_{p+1} M (r, f)}{log r},  (1.2)

where M (r, f) = max_{|z|=r} |f (z)| .

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

i (f) = \{ \begin{matrix}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. \end{matrix}  (1.3)

Definition 1.3 The iterated convergence exponent of the sequence of zeros of a meromorphic function f (z) is defined by

λ_{p}(f) = lim sup_{r→+∞} \frac{log_{p} N(r, 1/f)}{log r} (p ≥ 1 is an integer) ,  (1.4)

where N (r, \frac{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

\overline{λ}_{p}(f) = lim sup_{r→+∞} \frac{log_{p} \overline{N}(r, 1/f)}{log r} (p ≥ 1 is an integer) ,  (1.5)

where \overline{N} (r, \frac{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

i_{λ} (f) = \{ \begin{matrix}0,  if n(r, 1/f) = O (log r) , \\ min \{j ∈ ℕ : λ_{j} (f) < ∞\} , if λ_{j}(f) < ∞ for some j ∈ ℕ, \\ ∞,  if λ_{j}(f) = ∞ for all j ∈ ℕ. \end{matrix}

(1.6)

Remark 1.2 Similarly, we can define the finiteness degree i\overline{λ} (f) of \overline{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

f^{(n)} + A_{n−1} (z) f^{(n−1)} + ... + A_{1} (z) f′ + A_{0} (z) f = 0  (1.7)

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

|A_{0} (z)| ≥ exp {α |z|^{μ}}  (1.8)

and

|A_{j} (z)| ≤ exp {β |z|^{μ}} (j = 1, 2, ..., n − 1)  (1.9)

as z → ∞ in θ_{1} ≤ arg z ≤ θ_{2}. Then every solution f ≢ 0 of the equation (1.7) has an infinite order.

Theorem B [3] Let A_{0} (z) , ..., A_{n−1} (z) with A_{0} (z) ≢ 0 be entire functions. Suppose that there exist a sequence of complex numbers (z_{k})_{k∈ℕ} with lim_{k→+∞}z_{k} = ∞ and three real numbers α, β and μ satisfying 0 ≤ β < α and

μ > 0 such that

|A_{0} (z_{k})| ≥ exp {α |z_{k}|^{μ}}  (1.10)

and

|A_{j} (z_{k})| ≤ exp {β |z_{k}|^{μ}} (j = 1, 2, ..., n − 1)  (1.11)

as k → +∞. 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

A_{n} (z) f^{(n)} + A_{n−1} (z) f^{(n−1)} + ... + A_{1} (z) f′ + A_{0} (z) f = 0.  (1.12)

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

z^{2}f′′′ + 6zf′′ + 6f′ − z^{2}f = 0

has a meromorphic solution

f (z) = \frac{e^{z}}{z^{2}} .

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}) = 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

|A_{0} (z)| ≥ exp_{p} {α |z|^{σ−ε}}  (1.13)

and

|A_{j} (z)| ≤ exp_{p} {β |z|^{σ−ε}} (j = 1, 2, ..., n)  (1.14)

as z → ∞ in θ_{1} ≤ arg z ≤ θ_{2}. 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) = p + 1, σ_{p+1} (f) = σ.

Theorem 1.2 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}) = p (j = 0, 1, ..., n) and max {σ_{p} (A_{j}) : j = 1, 2, ..., n} < σ_{p} (A_{0}) = σ. Suppose that there exist a sequence of complex numbers (z_{k})_{k∈ℕ} with lim_{k→+∞}z_{k} = ∞

and two real numbers α and β satisfying 0 ≤ β < α such that for ε > 0 sufficiently small, we have

|A_{0} (z_{k})| ≥ exp_{p} {α |z_{k}|^{σ−ε}}  (1.15)

and

|A_{j} (z_{k})| ≤ exp_{p} {β |z_{k}|^{σ−ε}} (j = 1, 2, ..., n − 1)  (1.16)

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) = p + 1, σ_{p+1} (f) = σ.

Let A_{0} (z) , ..., A_{n−1} (z) , A_{n} (z) , F (z) be entire functions with A_{0} (z) ≢ 0, A_{n} (z) ≢ 0 and F ≢ 0. Considering the nonhomogeneous linear differential equation

A_{n} (z) f^{(n)} + A_{n−1} (z) f^{(n−1)} + ... + A_{1} (z) f′ + A_{0} (z) f = F,  (1.17)

we obtain the following result:

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.

(i) If q < p + 1 or q = p + 1 and σ_{p+1} (F) < σ_{p} (A_{0}) = σ, then every meromorphic solution f whose poles are of uniformly bounded multiplicity of the equation (1.17) satisfies i\overline{λ} (f) = i_{λ} (f) = i (f) = p + 1 and \overline{λ}_{p+1} (f) = λ_{p+1} (f) = σ_{p+1} (f) = σ with at most one exceptional solution f_{0} satisfying i (f_{0}) < p + 1 or σ_{p+1} (f_{0}) < σ. (ii) If q > p + 1 or q = p + 1 and σ_{p} (A_{0}) < σ_{p+1} (F) < +∞, then every meromorphic solution f whose poles are of uniformly bounded multiplicity of the equation (1.17) satisfies i (f) = q and σ_{q} (f) = σ_{q} (F).

2 Preliminary Lemmas

Lemma 2.1 [10] Let f (z) be a meromorphic function. Let α > 1 and Γ = {(k_{1}, j_{1}) , (k_{2}, j_{2}) , ..., (k_{m}, j_{m})} denote a set of distinct pairs of integers satisfying k_{i} > j_{i} ≥ 0. 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

|\frac{f^{(k)}(z)}{f^{(j)}(z)}| ≤ B [\frac{T (αr, f)}{r} (log^{α} r) log T (αr, f)]^{k−j} .  (2.1)

Lemma 2.2 [10] Let f (z) be a meromorphic function. Let α > 1 and ε > 0 be given constants. Then 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

|\frac{f^{(j)}(z)}{f(z)}| ≤ B [T (αr, f)r^{ε} log T (αr, f)]^{j} (j ∈ ℕ).  (2.2)

Lemma 2.3 [6, 7] Let p ≥ 1 be an integer and g(z) be an entire function with i (g) = p + 1 and σ_{p+1}(g) = σ. Let ν_{g}(r) be the central index of g (z) . Then

lim sup_{r→+∞} \frac{log_{p+1} ν_{g}(r)}{log r} = σ.  (2.3)

Lemma 2.4 Let p ≥ 1 be an integer and let f (z) = g (z) /d (z) be a meromorphic function, where g (z) and d (z) are entire functions satisfying σ_{p} (f) = σ_{p} (g) = +∞, i (d) < p or i (d) = p and σ_{p} (d) = ρ < +∞. Then there exist a sequence of complex numbers {z_{k}}_{k∈ℕ} and a set E_{3} of finite logarithmic measure such that |z_{k}| = r_{k} ∉ E_{3}, r_{k} → +∞, |g (z_{k})| = M (r_{k}, g) and for sufficiently large k, we have

\frac{f^{(n)}(z_{k})}{f(z_{k})} = (\frac{ν_{g} (r_{k})}{z_{k}})^{n} (1 + o (1)) (n ≥ 1 is an integer)  (2.4)

and

lim_{k→+∞} \frac{log_{p} ν_{g} (r_{k})}{log r_{k}} = σ_{p} (g) = +∞,  (2.5)

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

Proof. By induction, we obtain

f^{(n)} = \frac{g^{(n)}}{d} + ∑_{j=0}^{n−1} \frac{g^{(j)}}{d} ∑_{(j_{1}...j_{n})} C_{jj_{1}...j_{n}} (\frac{d′}{d} )^{j_{1}} ... (\frac{d^{(n)}}{d} )^{j_{n}} ,  (2.6)

where C_{jj_{1}...j_{n}} are constants and j + j_{1} + 2j_{2} + ... + nj_{n} = n. Hence

\frac{f^{(n)}}{f} = \frac{g^{(n)}}{g} + ∑_{j=0}^{n−1} \frac{g^{(j)}}{g} ∑_{(j_{1}...j_{n})} C_{jj_{1}...j_{n}} (\frac{d′}{d} )^{j_{1}} ... (\frac{d^{(n)}}{d} )^{j_{n}} .  (2.7)

From the Wiman-Valiron theory [12, 16], there exists a set E ⊂ (1, +∞) with finite logarithmic measure such that for a point z satisfying |z| = r ∉ E and |g (z)| = M (r, g), we have

\frac{g^{(j)}(z)}{g(z)} = (\frac{ν_{g} (r)}{z})^{j} (1 + o (1)) (j = 1, 2, ..., n) ,  (2.8)

where ν_{g} (r) is the central index of g. Substituting (2.8) into (2.7) yields

\frac{f^{(n)} (z)}{f (z)} = (\frac{ν_{g} (r)}{z})^{n} [(1 + o (1))

+ ∑_{j=0}^{n−1} (\frac{ν_{g} (r)}{z})^{j−n} (1 + o (1)) ∑_{(j_{1}...j_{n})} C_{jj_{1}...j_{n}} (\frac{d′}{d} )^{j_{1}} ... (\frac{d^{(n)}}{d} )^{j_{n}} ] . (2.9)

By Lemma 2.1, there exist a constant B > 0 and a set E_{1} ⊂ (1, +∞) having finite logarithmic measure such that for all z satisfying |z| = r ∉ [0, 1] ∪ E_{1}, we have

|\frac{d^{(m)} (z)}{d (z)}| ≤ B [T (2r, d)]^{2m} (m = 1, 2, ..., n) .  (2.10)

For any given ε > 0 and sufficiently large r, we have

T (2r, d) ≤ exp_{p−1} \{(2r)^{ρ+ \frac{ε}{2}} \} .  (2.11)

From (2.10) and (2.11) and j_{1}+2j_{2}+...+nj_{n} = n−j, we obtain for sufficiently large r, |z| = r ∉ [0, 1] ∪ E_{1}

|(\frac{d′}{d} )^{j_{1}} ... (\frac{d^{(n)}}{d} )^{j_{n}} | ≤ c [(exp_{p−1} \{(2r)^{ρ+ \frac{ε}{2}} \})^{2}]^{(n−j)}

= c [exp (2 exp_{p−2} \{(2r)^{ρ+ \frac{ε}{2}} \})]^{(n−j)} ≤ c [exp_{p−1} \{r^{ρ+ε}\}]^{(n−j)} , (2.12)

where c is a positive constant. Since σ_{p} (g) = +∞, it follows that there exists a sequence {r′_{k}} (r′_{k} → +∞) satisfying

lim_{_{k}→+∞} \frac{log_{p} ν_{g}(r′_{k})}{log r′_{k}} = +∞.  (2.13)

Setting the logarithmic measure of E_{3} = [0, 1] ∪ E ∪ E_{1}, lm (E_{3}) = δ < +∞, there exists a point r_{k} ∈ [r′_{k}, (δ + 1) r′_{k}] − E_{3}. Since

\frac{log_{p} ν_{g}(r_{k})}{log r_{k}} ≥ \frac{log_{p} ν_{g}(r′_{k})}{log [(δ + 1) r′_{k}]} = \frac{log_{p} ν_{g}(r′_{k})}{(log r′_{k}) [1 + \frac{log(δ+1)}{log r′_{k}}]},  (2.14)

we deduce that

lim_{_{k}→+∞} \frac{log_{p} ν_{g}(r_{k})}{log r_{k}} = +∞.  (2.15)

Then from (2.15) for a given arbitrary large L > ρ + ε + 1,

ν_{g} (r_{k}) > exp_{p−1} {r_{k}^{L}}  (2.16)

holds for sufficiently large r_{k}. This and (2.12) lead

|(\frac{ν_{g} (r_{k})}{z_{k}})^{j−n} (\frac{d′}{d} )^{j_{1}} ... (\frac{d^{(n)}}{d} )^{j_{n}} |

≤ c [\frac{r_{k} exp_{p−1} \{r_{k}^{ρ+ε}\}}{exp_{p−1} \{r_{k}^{L}\}}]^{(n−j)} → 0, r_{k} → +∞  (2.17)

for |z_{k}| = r_{k} and |g (z_{k})| = M (r_{k}, g) . From (2.15) , (2.9) and (2.17), we obtain our result.

Lemma 2.5 [6] Let p ≥ 1 be an integer. Suppose that f (z) is a meromorphic function such that i (f) = p, σ_{p} (f) = σ and i_{λ} (\frac{1}{f}) ≤ 1. Then for any given ε > 0, there exists a set E_{4} ⊂ (1, +∞) that has finite linear measure and finite logarithmic measure such that for all z satisfying |z| = r ∉ [0, 1] ∪ E_{4}, r → +∞, we have

|f (z)| ≤ exp_{p} {r^{σ+ε}} .  (2.18)

Lemma 2.6 [14] Let p ≥ 1 be an integer and let f (z) be a meromorphic function with i (f) = p. Then σ_{p} (f) = σ_{p} (f′).

Lemma 2.7 [6] Let p ≥ 1 be an integer and let f (z) be a meromorphic solution of the differential equation

f^{(n)} + B_{n−1} (z) f^{(n−1)} + ... + B_{1} (z) f′ + B_{0} (z) f = F,  (2.19)

where B_{0} (z) , ..., B_{n−1} (z) and F ≢ 0 are meromorphic functions such that (i) max {i (F) , i (B_{j}) (j = 0, ..., n − 1)} < i (f) = p + 1 or (ii) max {σ_{p+1} (F) , σ_{p+1} (B_{j}) (j = 0, ..., n − 1)} < σ_{p+1} (f). Then i\overline{λ} (f) = i_{λ} (f) = i (f) = p + 1 and \overline{λ}_{p+1} (f) = λ_{p+1} (f) = σ_{p+1} (f).

To avoid some problems caused by the exceptional set we recall the following lemmas.

Lemma 2.8 [1] Let g : [0, +∞) → ℝ and h : [0, +∞) → ℝ be monotone non-decreasing functions such that g (r) ≤ h (r) outside of an exceptional set E_{5} of finite linear measure. Then for any μ > 1, there exists r_{0} > 0 such that g (r) ≤ h (μr) for all r > r_{0}.

Lemma 2.9 [11] Let φ : [0, +∞) → ℝ and ψ : [0, +∞) → ℝ be monotone non-decreasing functions such that φ (r) ≤ ψ (r) for all r ∉ E_{6} ∪ [0, 1], where E_{6} ⊂ (1, +∞) is a set of finite logarithmic measure. Let η > 1 be a given constant. Then there exists an r_{1} = r_{1} (η) > 0 such that φ (r) ≤ ψ (ηr) for all r > r_{1}.

3 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

|A_{0} (z)| ≤ |A_{n} (z)| |\frac{f^{(n)}}{f} | + |A_{n−1} (z)| |\frac{f^{(n−1)}}{f} | + ... + |A_{1} (z)| |\frac{f′}{f} | . (3.1)

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

|\frac{f^{(j)}(z)}{f(z)}| ≤ Br [T (2r, f)]^{n+1} (j = 1, 2, ..., n) .  (3.2)

Hence from (1.13), (1.14), (3.1) and (3.2), it follows that

exp_{p} {α |z|^{σ−ε}} ≤ Bnr [T (2r, f)]^{n+1} exp_{p} {β |z|^{σ−ε}}  (3.3)

as r → +∞, |z| = r ∉ E_{2} and θ_{1} ≤ arg z ≤ θ_{2}. By Lemma 2.8 and (3.3), 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. Set

δ = max {σ_{p} (A_{j}) : j = 1, 2, ..., n} < σ_{p} (A_{0}) = σ < +∞.

We can rewrite (1.12) as

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

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} (\frac{1}{f}) ≤ p and λ_{p} (1/f) ≤ δ < σ < +∞. By Hadamard factorization theorem, we can write f as f (z) = g (z) /d (z), where g (z) and d (z) are entire functions satisfying i (f) = i (g) = t ≥ p + 1, σ_{t} (f) = σ_{t} (g) and i (d) ≤ p, σ_{p} (d) = λ_{p} (1/f) < σ < +∞. 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 |z_{k}| = r_{k} ∉ E_{3}, r_{k} → +∞, |g (z_{k})| = M (r_{k}, g) and for sufficiently large k, we have

\frac{f^{(j)}(z_{k})}{f(z_{k})} = (\frac{ν_{g} (r_{k})}{z_{k}})^{j} (1 + o (1)) (j = 1, 2, ..., n) .  (3.4)

By Remark 1.1, for any given ε > 0 and for sufficiently large r, we have

|A_{j} (z)| ≤ exp_{p} {r^{σ+ε}} (j = 0, 1, ..., n − 1) .  (3.5)

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 satisfying |z| = r ∉ [0, 1] ∪ E_{4}, r → +∞, we have

|1/A_{n} (z)| ≤ exp_{p} {r^{σ+ε}} .  (3.6)

We can rewrite (1.12) as

−A_{n} (z) \frac{f^{(n)}}{f} = A_{n−1} (z) \frac{f^{(n−1)}}{f} + ... + A_{1} (z) \frac{f′}{f} + A_{0} (z) .  (3.7)

Substituting (3.4) into (3.7), we obtain for the above z_{k}

−A_{n} (z_{k}) (\frac{ν_{g} (r_{k})}{z_{k}})^{n} (1 + o (1)) = A_{n−1} (z_{k}) (\frac{ν_{g} (r_{k})}{z_{k}})^{n−1} (1 + o (1))

+... + A_{1} (z_{k}) (\frac{ν_{g} (r_{k})}{z_{k}}) (1 + o (1)) + A_{0} (z_{k}) .  (3.8)

Hence from (3.5), (3.6) and (3.8), we have

(1/ exp_{p} \{r_{k}^{σ+ε}\}) |\frac{ν_{g} (r_{k})}{z_{k}} |^{n} |1 + o (1)|

≤ exp_{p} \{r_{k}^{σ+ε}\} |\frac{ν_{g} (r_{k})}{z_{k}} |^{n−1} |1 + o (1)|

+... + exp_{p} \{r_{k}^{σ+ε}\} |\frac{ν_{g} (r_{k})}{z_{k}} | |1 + o (1)| + exp_{p} \{r_{k}^{σ+ε}\}

≤ n exp_{p} \{r_{k}^{σ+ε}\} |\frac{ν_{g} (r_{k})}{z_{k}} |^{n−1} |1 + o (1)| ,  (3.9)

where |z_{k}| = r_{k} ∉ [0, 1] ∪ E_{3} ∪ E_{4}, r_{k} → +∞ and |g (z_{k})| = M (r_{k}, g). By Lemma 2.9 and (3.9), we get

lim sup_{k→+∞} \frac{log_{p+1} ν_{g}(r_{k})}{log r_{k}} ≤ σ + ε.  (3.10)

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 σ_{p+1} (f) ≥ σ yield i (f) = p + 1 and σ_{p+1} (f) = σ.

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

|A_{0} (z)| ≤ |A_{n} (z)| |\frac{f^{(n)}}{f} | + |A_{n−1} (z)| |\frac{f^{(n−1)}}{f} | + ... + |A_{1} (z)| |\frac{f′}{f} | . (4.1)

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

|\frac{f^{(j)}(z)}{f(z)}| ≤ Br [T (2r, f)]^{n+1} (j = 1, 2, ..., n) .  (4.2)

Hence from (1.15), (1.16), (4.1) and (4.2), we have

exp_{p} {α |z_{k}|^{σ−ε}} ≤ Bnr_{k} [T (2r_{k}, f)]^{n+1} exp_{p} {β |z_{k}|^{σ−ε}}  (4.3)

as k → +∞, |z_{k}| = r_{k} ∉ E_{2}. 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.

5 Proof of Theorem 1.3

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

f (z) = B_{1} (z) f_{1} (z) + B_{2} (z) f_{2} (z) + ... + B_{n} (z) f_{n} (z) ,  (5.1)

where B_{1} (z) , ..., B_{n} (z) are suitable meromorphic functions determined by

\begin{matrix}B′_{1} (z) f_{1} (z) + B′_{2} (z) f_{2} (z) + ... + B′_{n} (z) f_{n} (z) = 0 \\ B′_{1} (z) f′_{1} (z) + B′_{2} (z) f′_{2} (z) + ... + B′_{n} (z) f′_{n} (z) = 0 \\ ............................... \\ B′_{1} (z) f_{1}^{(n−1)} (z) + B′_{2} (z) f_{2}^{(n−1)} (z) + ... + B′_{n} (z) f_{n}^{(n−1)} (z) = F (z) . \end{matrix} (5.2)

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.  (5.3)

From (5.2), we get

B′_{j} = F.G_{j} (f_{1}, f_{2}, ..., f_{n}) .W (f_{1}, f_{2}, ..., f_{n})^{−1} (j = 1, 2, ..., n) ,  (5.4)

where G_{j} (f_{1}, f_{2}, ...f_{n}) are differential polynomials in f_{1}, f_{2}, ..., f_{n} with constant coefficients. Thus

σ_{p+1} (G_{j}) ≤ max {σ_{p+1} (f_{j}) : j = 1, 2, ..., n}
= σ_{p} (A_{0}) = σ (j = 1, 2, ..., n) .  (5.5)

(i) If q < p + 1 or q = p + 1 and σ_{p+1} (F) < σ_{p} (A_{0}) = σ, then by Lemma 2.6, (5.3) , (5.4) and (5.5) for j = 1, 2, ..., n, we have

σ_{p+1} (B_{j}) = σ_{p+1} (B′_{j}) ≤ max {σ_{p+1} (F) , σ_{p} (A_{0})} = σ_{p} (A_{0}) = σ. (5.6)

Then from (5.1) and (5.6), we get

σ_{p+1} (f) ≤ max {σ_{p+1} (f_{j}) , σ_{p+1} (B_{j}) : j = 1, 2, ..., n}
= σ_{p} (A_{0}) = σ < +∞.  (5.7)

From (1.17), it follows that

|A_{0} (z)| ≤ |A_{n} (z)| |\frac{f^{(n)}}{f} |+|A_{n−1} (z)| |\frac{f^{(n−1)}}{f} |+...+|A_{1} (z)| |\frac{f′}{f} |+|\frac{F}{f} | . (5.8)

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

|\frac{f^{(j)}(z)}{f(z)}| ≤ Br [T (2r, f)]^{n+1} (j = 1, 2, ..., n) .  (5.9)

Set

max {σ_{p} (A_{j}) : j = 1, 2, ..., n} = δ < σ_{p} (A_{0}) = σ.  (5.10)

We can rewrite (1.17) as

f^{(n)} + \frac{A_{n−1} (z)}{A_{n} (z)}f^{(n−1)} + ... + \frac{A_{1} (z)}{A_{n} (z)}f′ + \frac{A_{0} (z)}{A_{n} (z)}f = \frac{F}{A_{n} (z)}.  (5.11)

Obviously, it follows that 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} (\frac{1}{f}) ≤ p and λ_{p} (1/f) ≤ δ < σ < +∞. By Hadamard factorization theorem, we can write f as f (z) = g (z) /d (z), where g (z) and d (z) are entire functions satisfying i (f) = i (g) = t ≥ p + 1, σ_{t} (f) = σ_{t} (g) and i (d) ≤ p, σ_{p} (d) = λ_{p} (1/f) < σ < +∞. Set

max {σ_{p+1} (F) , σ_{p} (d)} = γ < σ.  (5.12)

For any given ε (0 < 2ε < σ − γ) and a sufficiently large r, we have

|F (z)| ≤ exp_{p} {r^{γ+ε}} and |d (z)| ≤ exp_{p−1} {r^{γ+ε}} .  (5.13)

Since M (r, g) ≥ 1 for a sufficiently large r, we obtain from (5.13),

|\frac{F (z)}{f (z)}| = \frac{|F (z)| |d (z)|}{|g (z)|} ≤ exp_{p} \{r^{γ+ε}\} exp_{p−1} \{r^{γ+ε}\}  (5.14)

as r → +∞, |z| = r and |g (z)| = M (r, g). If A_{0} (z) , ..., A_{n−1} (z) and A_{n} (z) satisfy the hypotheses of Theorem 1.2, then from (1.15), (1.16) , (5.8) , (5.9) and (5.14), it follows that

exp_{p} {α |z_{k}|^{σ−ε}} ≤ Bnr_{k} [T (2r_{k}, f)]^{n+1} exp_{p} {β |z_{k}|^{σ−ε}}
+ exp_{p} {|z_{k}|^{γ+ε}} exp_{p−1} {|z_{k}|^{γ+ε}}  (5.15)

as k → +∞, |z_{k}| = r_{k} ∉ E_{2} and |g (z_{k})| = M (r_{k}, g). From (5.15) and Lemma 2.8, we get σ_{p+1} (f) ≥ σ − ε. Since ε > 0 is arbitrary, it follows that σ_{p+1} (f) ≥ σ. This and the fact that σ_{p+1} (f) ≤ σ yield σ_{p+1} (f) = σ. Thus by Lemma 2.7, we have i\overline{λ} (f) = i_{λ} (f) = i (f) = p + 1 and \overline{λ}_{p+1} (f) = λ_{p+1} (f) = σ_{p+1} (f) = σ.

(ii) If q > p + 1 or q = p + 1 and σ_{p} (A_{0}) < σ_{p+1} (F) < +∞, then by Lemma 2.6, (5.3) , (5.4) and (5.5) for j = 1, 2, ..., n, we have

σ_{q} (B_{j}) = σ_{q} (B′_{j}) ≤ max {σ_{q} (F) , σ_{q} (f_{j}) : j = 1, 2, ..., n} = σ_{q} (F) . (5.16)

Then from (5.1) and (5.16), we get

σ_{q} (f) ≤ max {σ_{q} (f_{j}) , σ_{q} (B_{j}) : j = 1, 2, ..., n} = σ_{q} (F) .  (5.17)

On the other hand, if q > p + 1 or q = p + 1 and σ_{p} (A_{0}) < σ_{p+1} (F) < +∞, it follows from (1.17) that a simple consideration of order implies σ_{q} (f) ≥ σ_{q} (F). By this inequality and (5.17) we obtain σ_{q} (f) = σ_{q} (F) .

References

[1] S. Bank, General theorem concerning the growth of solutions of firstorder algebraic differential equations, Compositio Math. 25 (1972), 61– 70.

[2] B. Belaïdi and K. Hamani, Order and hyper-order of entire solutions of linear differential equations with entire coefficients, Electron. J. Differential Equations 2003, No. 17, 1-12.

[3] B. Belaïdi and S. Hamouda, Growth of solutions of n-th order linear differential equation with entire coefficients, Kodai Math. J. 25 (2002), 240-245.

[4] B. Belaïdi and S. Hamouda, Orders of solutions of an n-th order linear differential equations with entire coefficients, Electron. J. Differential Equations 2001, No. 61, 1-5.

[5] L. G. Bernal, On growth k-order of solutions of a complex homogeneous linear differential equation, Proc. Amer. Math. Soc. 101 (1987), no. 2, 317–322.

[6] T. B. Cao and H. X. Yi, On the complex oscillation of higher order linear differential equations with meromorphic coefficients, J. Syst. Sci. Complex. 20 (2007), no. 1, 135–148.

[7] 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.

[8] Z. X. Chen and C. C. Yang, Some further results on the zeros and growths of entire solution of second order linear differential equations, Kodai Math. J., 22 (1999), 273-285.

[9] W. J. Chen and J. F. Xu, Growth of meromorphic solutions of higherorder linear differential equations, Electron. J. Qual. Theory Differ. Equ. 2009, No. 1, 1-13.

[10] G. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), no. 1, 88–104.

[11] G. G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), no. 1, 415–429.

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

[13] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs Clarendon Press, Oxford, 1964.

[14] L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math. 22 (1998), no. 4, 385–405.

[15] I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Mathematics, 15. Walter de Gruyter & Co., Berlin, 1993.

[16] G. Valiron, Lecture on the general theory of integral functions, Chelsea, New York, 1949.

[17] J. Xu and Z. Zhang, Growth order of meromorphic solutions of higher order linear differential equations, Kyungpook Math. J., 48 (2008), 123-132.

[18] L. Z. Yang, The growth of linear differential equations and their applications, Israel J. Math., 147 (2005), 359-370.

(Received December 15, 2009)