PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 199–210 DOI: 10.2298/PIM141009007A

ON THE GROWTH AND THE ZEROS OF SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

WITH MEROMORPHIC COEFFICIENTS

Maamar Andasmas and Benharrat Belaïdi

Abstract. We investigate the growth of meromorphic solutions of homogeneous and nonhomogeneous higher order linear differential equations

f^{(k)} + ∑_{j=1}^{k−1} A_{j} f^{(j)} + A_{0}f = 0 (k ⩾ 2),

f^{(k)} + ∑_{j=1}^{k−1} A_{j} f^{(j)} + A_{0}f = A_{k} (k ⩾ 2),

where A_{j} (z) (j = 0, 1, …, k) are meromorphic functions with finite order. Under some conditions on the coefficients, we show that all meromorphic solutions f ≢ 0 of the above equations have an infinite order and infinite lower order. Furthermore, we give some estimates of their hyper-order, exponent and hyper-exponent of convergence of distinct zeros. We improve the results due to Kwon; Chen and Yang; Belaïdi; Chen; Shen and Xu.

1. Introduction and statement of results

In this paper, we use the standard notations of Nevanlinna’s value distribution theory [10, 12, 15]. In addition, we use the notations λ(f) and λ(1/f) to denote respectively the exponents of convergence of the zeros and the poles of a meromorphic function f, ρ(f) and μ(f) to denote respectively the order and the lower order of f. The hyper-order ρ_{2}(f), the hyper-exponent λ_{2}(f) of convergence of zeros and the hyper-exponent \overline{λ}_{2}(f) of convergence of distinct zeros of f are defined respectively by

ρ_{2}(f) = lim sup_{r→+∞} \frac{loglogT (r, f)}{logr} , λ_{2}(f) = lim sup_{r→+∞} \frac{loglogN(r, 1/f)}{logr} ,

2010 Mathematics Subject Classification: Primary 34M10; Secondary 30D35. Key words and phrases: linear differential equations; meromorphic functions; order of growth;

hyper-order; exponent of convergence of zeros; hyper-exponent of convergence of zeros.

Partially supported by University of Mostaganem (UMAB) (CNEPRU Project Code B02220120024).

Communicated by Gradimir Milovanović.

\overline{λ}_{2}(f) = lim sup_{r→+∞} \frac{loglog \overline{N}(r, 1/f)}{logr} .

First, we recall the following definitions. The linear measure of a set E ⊂ (0, +∞) is defined as m(E) = ∫_{0}^{+∞} χ_{E}(t) dt and the logarithmic measure of a set F ⊂ (1, +∞) is defined by lm(F) = ∫_{1}^{+∞} \frac{1}{t}χ_{F} (t) dt, where χ_{H} (t) is the characteristic function of a set H. The upper density of a set E ⊂ (0, +∞) is defined by

\overline{dens}E = lim sup_{r→+∞} \frac{m(E ∩ [0, r])}{r} .

The upper logarithmic density of a set F ⊂ (1, +∞) is defined by

\overline{log dens}(F) = lim sup_{r→+∞} \frac{lm(F ∩ [1, r])}{logr}.

Proposition 1.1. For all H ⊂ [1, +∞) the following statements hold:

i) If lm(H) = ∞, then m(H) = ∞; ii) If \overline{dens}H > 0, then m(H) = ∞; iii) If \overline{log dens}H > 0, then lm(H) = ∞.

Proof. i) Since we have χ_{H} (t)/t ⩽ χ_{H} (t) for all t ∈ H ⊂ [1, +∞), then

m(H) ⩾ lm(H).

We can easily prove the results ii) and iii) by applying the definition of the limit and the properties m(H ∩ [0, r]) ⩽ m(H) and lm(H ∩ [1, r]) ⩽ lm(H). □

In [11], Kwon investigated the growth of second order equations and obtained the following result.

Theorem 1.1. [11] Let H be a set of complex numbers satisfying \overline{dens}\{|z| : z ∈ H} > 0, and let A(z) and B(z) be entire functions such that for real constants α (> 0), β (> 0),

|A(z)| ⩽ exp{o(1)|z|^{β}} and |B(z)| ⩾ exp{(1 + o(1))α|z|^{β}}

as z → ∞ for z ∈ H. Then every solution f ≢ 0 of the equation

(1.1)  f′′+ A(z)f′+ B(z)f = 0

has infinite order and ρ_{2}(f) ⩾ β.

In [6], Chen and Yang have studied the growth of solutions of (1.1) and obtained the following result.

Theorem 1.2. [6] Let H be a set of complex numbers satisfying \overline{dens}\{|z| : z ∈ H} > 0, and let A(z) and B(z) be entire functions with ρ(A) ⩽ ρ(B) = ρ < +∞ such that for real constant C (> 0) and for any given ε > 0,

|A(z)| ⩽ exp{o(1)|z|^{ρ−ε} } and |B(z)| ⩾ exp{(1 + o(1))C|z|^{ρ−ε} }

as z → ∞ for z ∈ H. Then every solution f ≢ 0 of equation (1.1) has infinite order and ρ_{2}(f) = ρ(B).

These results were improved by Belaïdi in [1, 2] by considering more general conditions to higher order linear differential equations with entire coefficients. Recently in [8], Chen extended the previous results by studying the zeros and the growth of meromorphic solutions of equation (1.1) and the non-homogeneous equation f′′+ A(z)f′+ B(z)f = F when A(z), B(z), F (z) are meromorphic functions.

Here we consider for k ⩾ 2 the homogeneous and the non-homogeneous linear differential equations

f^{(k)} + ∑_{j=1}^{k−1} A_{j} f^{(j)} + A_{0}f = 0, (1.2)

f^{(k)} + ∑_{j=1}^{k−1} A_{j} f^{(j)} + A_{0}f = A_{k}, (1.3)

where A_{j} (z) (j = 0, 1, …, k) (A_{0} ≢ 0 and A_{k} ≢ 0) are meromorphic functions with finite order. We investigate the zeros and growth of meromorphic solutions of equations (1.2) and (1.3). The present article may be understood as an extension and improvement of Theorems 2.3 and 2.4 in the paper of Shen and Xu [14]. We improve the results due to Chen; Shen and Xu greatly and we give two corollaries in the case when ρ = max{ρ(A_{j}) : j = 1, 2, …, k} < ρ(A_{0}) = σ < 1/2.

Theorem 1.3. Let H ⊂ [0, +∞) be a set with infinite linear measure, and let A_{j} (z) (j = 0, 1, …, k −1) be meromorphic functions with finite order. If there exist positive constants σ > 0, α > 0 such that ρ = max{ρ(A_{j}) : j = 1, 2, …, k − 1} < σ and |A_{0}(z)| ⩾ e^{αr^{σ}} as |z| = r ∈ H, r → +∞, then every meromorphic solution f ≢ 0 of equation (1.2) satisfies μ(f) = ρ(f) = ∞ and ρ_{2}(f) ⩾ σ. Furthermore, if λ(1/f) < ∞, then σ ⩽ ρ_{2}(f) ⩽ ρ(A_{0}).

Remark 1.1. By using Proposition 1.1, we can obtain the same results in Theorem 1.3, while putting H ⊂ [0, +∞) to be a set with positive upper density (or while putting H ⊂ [1, +∞) to be a set with positive upper logarithmic density) (or while putting H ⊂ [1, +∞) to be a set with infinite logarithmic measure) instead of putting H to be a set with infinite linear measure.

Theorem 1.4. Let H ⊂ [0, +∞) be a set with a positive upper density, and let A_{j} (z) (j = 0, 1, …, k) (A_{k} ≢ 0) be meromorphic functions with finite order. If there exist positive constants σ > 0, α > 0 such that ρ = max{ρ(A_{j}) : j = 1, 2, …, k} < σ and |A_{0}(z)| ⩾ e^{αr^{σ}} as |z| = r ∈ H, r → +∞, then every meromorphic solution f with λ(1/f) < σ of equation (1.3) is of infinite order and

\overline{λ}(f) = λ(f) = ρ(f) = ∞, \overline{λ}_{2}(f) = λ_{2}(f) = ρ_{2}(f).

Furthermore, if λ(1/f) < min{μ(f), σ}, then ρ_{2}(f) ⩽ ρ(A_{0}).

Remark 1.2. It is clear that ρ(A_{0}) = β ⩾ σ in Theorems 1.3 and 1.4. Indeed, suppose that ρ(A_{0}) = β < σ. Then, by using Lemma 2.2 of this paper, there exists a set E_{1} ⊂ (1, +∞) that has a finite linear measure such that when |z| = r ∉ [0, 1] ∪ E_{1}, r → +∞, we have for any given ε (0 < ε < σ − β)

(1.4)  |A_{0}(z)| ⩽ e^{r^{β+ε}} .

On the other hand, by the hypotheses of Theorems 1.3 and 1.4, there exist positive constants σ > 0, α > 0 such that

(1.5)  |A_{0}(z)| ⩾ e^{αr^{σ}}

as |z| = r ∈ H, r → +∞, where H is a set with m(H) = ∞. From (1.4) and (1.5), we obtain for |z| = r ∈ H ∖ [0, 1] ∪ E_{1}, r → +∞

e^{αr^{σ}} ⩽ |A_{0}(z)| ⩽ e^{r^{β+ε}}

and by ε (0 < ε < σ − β) this is a contradiction as r → +∞. Hence ρ(A_{0}) = β ⩾ σ.

Corollary 1.1. Let A_{j} (z) (j = 0, 1, …, k) (A_{k} ≢ 0) be meromorphic func-

tions having only finitely many poles such that ρ = max{ρ(A_{j}) : j = 1, 2, …, k} < ρ(A_{0}) = σ < 1/2. Then every meromorphic solution f ≢ 0 of equation (1.2) satisfies μ(f) = ρ(f) = ∞ and ρ_{2}(f) = σ. Furthermore, every meromorphic solution fof equation (1.3) satisfies

\overline{λ}(f) = λ(f) = ρ(f) = ∞, \overline{λ}_{2}(f) = λ_{2}(f) = ρ_{2}(f)

and if λ(1/f) < μ(f), then ρ_{2}(f) ⩽ ρ(A_{0}).

Corollary 1.2. Let A_{j} (z) (j = 0, 1, …, k) (A_{k} ≢ 0) be meromorphic func-

tions such that λ(1/A_{0}) < μ(A_{0}) ⩽ ρ(A_{0}) = σ < 1/2 and ρ = max{ρ(A_{j}) : j = 1, 2, …, k} < σ. Then every meromorphic solution f ≢ 0 of equation (1.2) whose all poles are of uniformly bounded multiplicity satisfies μ(f) = ρ(f) = ∞ and ρ_{2}(f) = σ. Furthermore, every meromorphic solution f of equation (1.3) whose all poles are of uniformly bounded multiplicity satisfies

\overline{λ}(f) = λ(f) = ρ(f) = ∞, \overline{λ}_{2}(f) = λ_{2}(f) = ρ_{2}(f)

and if λ(1/f) < μ(f), then ρ_{2}(f) ⩽ ρ(A_{0}).

2. Auxiliary lemmas

Lemma 2.1. [9] Let f(z) be a transcendental meromorphic function, and let α > 1, ε > 0 be given constants. Then there exists a set E_{0} ⊂ [0, ∞) that has finite linear measure and there exists a constant c > 0, such that for all z satisfying |z| = r ∉ E_{0}, we have

| \frac{f^{(j)} (z)}{f(z)} | ⩽ c[T (αr, f)r^{ε} logT (αr, f)]^{j} (j ∈ ℕ).

Lemma 2.2. [5] Let f(z) be a meromorphic function of order ρ(f) = ρ < +∞. Then for any given ε > 0, there exists a set E_{1} ⊂ (1, +∞) that has finite linear measure and finite logarithmic measure such that when |z| = r ∉ [0, 1] ∪ E_{1}, r → +∞ , we have |f(z)| ⩽ exp{r^{ρ+ε} }.

Let g(z) = ∑ _{n=0}^{∞} a_{n} z^{n} be an entire function. We define by μ(r) = max{|a_{n}|r^{n}; n = 0, 1, ⋯ } the maximum term of g, and define by ν_{g}(r) = max{m; μ(r) = |a_{m}|r^{m}} the central index of g.

Lemma 2.3. [7] Let f(z) = g(z)/d(z) be a meromorphic function, where g(z) and d(z) are entire functions such that

μ(g) = μ(f) = μ ⩽ ρ(g) = ρ(f) ⩽ +∞,
λ(d) = ρ(d) = λ(1/f) = β < μ.

Let z be a point with |z| = r at which |g(z)| = M(r, g) and ν_{g}(r) denote the central index of g(z). Then there exists a set E_{2} ⊂ (1, +∞) with finite logarithmic measure lm(E_{2}) < ∞, such that for all z satisfying |z| = r ∉ [0, 1] ∪ E_{2}, we have

\frac{f^{(n)} (z)}{f(z)} = ( \frac{ν_{g}(r)}{z})^{n}(1 + o(1)) (n ⩾ 1 is an integer).

Lemma 2.4. [6] Let g(z) be an entire function of infinite order, with the hyperorder ρ_{2}(g) = σ. Then

lim sup_{r→+∞} \frac{loglogν_{g}(r)}{logr} = σ.

Lemma 2.5. [13] Let g(z) be an entire function of infinite order. Denote M(r, g) = max{|g(z)| : |z| = r}, then for any sufficiently large number λ > 0, and any r ∈ H_{0} ⊂ (1, +∞)

M(r, g) > c_{1} exp{c_{2}r^{λ}},

where lm(H_{0}) = ∞ and c_{1}, c_{2} are positive constants.

Lemma 2.6. Suppose that k ⩾ 2 and h_{0}, h_{1}, …, h_{k} (h_{0} ≢ 0) are meromorphic functions. Let ρ = max{ρ(h_{j}) : j = 0, 1, …, k} < ∞ and let f be a meromorphic solution of infinite order of the equation

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

with λ(1/f) = μ < μ(f). Then ρ_{2}(f) ⩽ ρ.

Proof. We assume that f is a meromorphic solution of infinite order ρ(f) = ∞ of equation (2.1). We can rewrite (2.1) as

(2.2)  | \frac{f^{(k)}}{f} | ⩽ |h_{0}| + ∑_{s=1}^{k−1} |h_{s}|| \frac{f^{(s)}}{f} | + | \frac{h_{k}}{f} |.

By Hadamard factorization theorem, we can write f as f(z) = \frac{g(z)}{d(z)} , where g(z) and d(z) are entire functions such that

μ(f) = μ(g) ⩽ ρ(f) = ρ(g) = ∞, λ(d) = ρ(d) = λ(1/f) = μ < μ(f)
ρ_{2}(f) = ρ_{2}(g).

By Lemma 2.3, there exists a set E_{2} ⊂ (1, +∞) with finite logarithmic measure, such that for all z satisfying |z| = r ∉ [0, 1] ∪ E_{2} at which |g(z)| = M(r, g), we have

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

Since ρ(h_{k}(z)d(z)) ⩽ ρ_{1} = max{μ, ρ}, then by Lemma 2.2, for any ε > 0 there exists a set E_{1} ⊂ (1, +∞) with a finite logarithmic measure such that

(2.4)  |h_{k}(z)d(z)| ⩽ e^{r^{ρ_{1} +ε}}

holds for all z satisfying |z| = r ∉ [0, 1] ∪ E_{1}, r → +∞. On the other hand, by Lemma 2.5 for a sufficiently large number λ > ρ_{1}, there exist a set H_{0} ⊂ (1, +∞) with lm(H_{0}) = ∞ and positive constants c_{1}, c_{2} such that for any r ∈ H_{0}

(2.5)  M(r, g) > c_{1} exp{c_{2}r^{λ}}.

By (2.4) and (2.5), for any given ε with 0 < ε < λ − ρ_{1} and for all z satisfying |z| = r ∈ H_{0} ∖ ([0, 1] ∪ E_{1}), r → +∞ at which |g(z)| = M(r, g), we have

(2.6)  | \frac{h_{k}(z)}{f(z)} | = | \frac{d(z)h_{k}(z)}{g(z)} | ⩽ \frac{e^{r^{ρ_{1} +ε}}}{c_{1}e^{c_{2} r^{λ}}} → 0, r → +∞.

Since ρ = max{ρ (h_{j}) : j = 0, 1, …, k} < ∞, then by Lemma 2.2, we have

(2.7)  |h_{j} (z)| ⩽ e^{r^{ρ+ε}} (j = 0, 1, …, k)

holds for all z satisfying |z| = r ∉ [0, 1] ∪ E_{1}, r → +∞. Substituting (2.3), (2.6) and (2.7) into (2.2), we obtain for all z satisfying |z| = r ∈ H_{0} ∖ ([0, 1] ∪ E_{1} ∪ E_{2}), r → +∞ at which |g(z)| = M(r, g),

| \frac{ν_{g}(r)}{z} |^{k}|1 + o(1)| ⩽ e^{r^{ρ+ε}} + ∑_{s=1}^{k−1} e^{r^{ρ+ε}} | \frac{ν_{g}(r)}{z} |^{s}|1 + o(1)| + o(1).

So, we get

(2.8)  |ν_{g}(r)|^{k}|1 + o(1)| ⩽ (k + 1)r^{k}e^{r^{ρ+ε}} |ν_{g}(r)|^{k−1} |1 + o(1)|.

Then, by Lemma 2.4, we obtain from (2.8) that ρ_{2}(g) = ρ_{2}(f) ⩽ ρ + ε. Since ε (0 < ε < λ − ρ_{1}) being arbitrary, then we get ρ_{2}(f) ⩽ ρ. □

Lemma 2.7. [3] Let f(z) be an entire function of order ρ, where 0 < ρ(f) = ρ < 1/2, and let ε > 0 be a given constant. Then there exists a set H_{1} ⊂ [0, +∞) with \overline{dens}H_{1} ⩾ 1−2ρ such that |f(z)| ⩾ exp\{r^{ρ−ε} \} for all z satisfying |z| = r ∈ H_{1}.

Lemma 2.8. [7] Suppose that h(z) is a meromorphic function with λ(1/h) < μ(h) ⩽ ρ(h) = σ < 1/2. Then for any ε > 0, there exists a set H_{2} ⊂ (1, +∞) that has a positive upper logarithmic density such that for all z satisfying |z| = r ∈ H_{2}, we have |h(z)| ⩾ exp{(1 + o(1))r^{σ−ε} }.

Lemma 2.9. [11] Let g(z) be a nonconstant entire function of finite order. Then for any given ε > 0, there exists a set H_{3} ⊂ [0, +∞) with \overline{dens}H_{3} = 1 such that M(r, g) ⩾ exp{r^{ρ(g)−ε} } for all z satisfying |z| = r ∈ H_{3}.

Lemma 2.10. [4] Let A_{j} (j = 0, 1, …, k−1), F ≢ 0 be finite order meromorphic functions. If f(z) is an infinite order meromorphic solution of the equation

f^{(k)} + A_{k−1} f^{(k−1)} + ⋯+ A_{1}f′+ A_{0}f = F,

then f satisfies \overline{λ}(f) = λ(f) = ρ(f) = ∞.

Lemma 2.11. Let H ⊂ [0, +∞) be a set with a positive upper density (or of infinite linear measure), and let h_{j} (z) (j = 0, 1, …, k) (h_{0} ≢ 0) be meromorphic functions with finite order. If there exist positive constants σ > 0, α > 0 such that |h_{0}(z)| ⩾ e^{αr^{σ}} as |z| = r ∈ H, r → +∞, and ρ = max{ρ(h_{j}) : j = 1, 2, …, k} < σ, then every meromorphic solution f ≢ 0 of equation

(2.9)  f^{(k)} + ∑_{j=1}^{k−1} h_{j} f^{(j)} + h_{0}f = h_{k} (k ⩾ 2),

is transcendental and satisfies ρ(f) ⩾ σ.

Proof. Assume that f ≢ 0 is a meromorphic solution of (2.9) with ρ(f) < σ. It follows from (2.9) that

(2.10)  \frac{h_{k}}{f} − \frac{f^{(k)}}{f} − ∑_{j=1}^{k−1} h_{j} \frac{f^{(j)}}{f} = h_{0}.

Since ρ(h_{j}) < σ (j = 1, 2, …, k) and ρ(f) < σ, then from (2.10) we obtain that the order of growth of h_{0} is ρ_{1} = ρ(h_{0}) ⩽ max{ρ, ρ(f)} < σ. By Lemma 2.2, for any ε (0 < ε < σ − ρ_{1}) there exists a set E_{1} ⊂ (1, +∞) with a finite linear measure such that

(2.11)  |h_{0}(z)| ⩽ e^{r^{ρ_{1} +ε}}

holds for all z satisfying |z| = r ∉ [0, 1] ∪ E_{1}, r → +∞. From the hypotheses of Lemma 2.11, there exists a set H with \overline{dens}H > 0 (or m(H) = ∞), and there exist positive constants σ > 0, α > 0 such that

(2.12)  |h_{0}(z)| ⩾ e^{αr^{σ}}

holds for all z satisfying |z| = r ∈ H, r → +∞. By (2.11) and (2.12), we conclude that for all z satisfying |z| = r ∈ H ∖ ([0, 1] ∪ E_{1}), r → +∞, we have e^{αr^{σ}} ⩽ e^{r^{ρ_{1} +ε}} and by ε (0 < ε < σ − ρ_{1}) this is a contradiction as r → +∞. Consequently, any meromorphic solution f ≢ 0 of equation (2.9) is transcendental and satisfies ρ(f) ⩾ σ. □

3. Proofs of the results

Proof of Theorem 1.3. Let f ≢ 0 be a meromorphic solution of (1.2). It follows from (1.2) that

(3.1)  |A_{0}| ⩽ | \frac{f^{(k)}}{f} | + ∑_{j=1}^{k−1} |A_{j} || \frac{f^{(j)}}{f} |.

By Lemma 2.11, we know that f is transcendental. By using Lemma 2.1, there is a set E_{0} ⊂ (0, +∞) having finite linear measure such that for all z satisfying |z| = r ∉ E_{0}, we have

(3.2)  | \frac{f^{(j)} (z)}{f(z)} | ⩽ r[T (2r, f)]^{2k} (j = 1, 2, …, k).

By Lemma 2.2, for any given ε (0 < ε < σ − ρ) there exists a set E_{1} ⊂ (1, +∞) with finite linear measure such that

(3.3)  |A_{j} (z)| ⩽ e^{r^{ρ+ε}} , j = 1, 2, …, k − 1

holds for all z satisfying |z| = r ∉ [0, 1] ∪ E_{1}, r → +∞. Also, by the hypotheses of Theorem 1.3, there exists a set H with m(H) = ∞, such that for all z satisfying |z| = r ∈ H, r → +∞, we have

(3.4)  |A_{0}(z)| ⩾ e^{αr^{σ}} .

Hence it follows from (3.1), (3.2), (3.3) and (3.4) that for all z satisfying |z| = r ∈ H ∖ ([0, 1] ∪ E_{0} ∪ E_{1}), r → +∞, we have

(3.5)  e^{αr^{σ}} ⩽ r[T (2r, f)]^{2k} + ∑_{j=1}^{k−1} e^{r^{ρ+ε}} r[T (2r, f)]^{2k} ⩽ kre^{r^{ρ+ε}} [T (2r, f)]^{2k}.

By 0 < ε < σ − ρ, it follows from (3.5) that

(3.6)  μ(f) = ρ(f) = ∞ and ρ_{2}(f) ⩾ σ.

Furthermore, if λ(1/f) < ∞, then f is a meromorphic solution of (1.2) with ρ(f) = μ(f) = ∞, λ(1/f) < μ(f) and by Remark 1.2, we have max{ρ(A_{j}) : j = 0, 1, …, k − 1 } = ρ(A_{0}) = β < ∞. Thus, by Lemma 2.6, we get

(3.7)  ρ_{2}(f) ⩽ ρ(A_{0}).

By (3.6) and (3.7), we conclude that σ ⩽ ρ_{2}(f) ⩽ ρ(A_{0}). □

Proof of Theorem 1.4. Let f be a meromorphic solution of (1.3). Assume that ρ(f) < ∞. It follows from (1.3) that

(3.8)  |A_{0}| ⩽ | \frac{f^{(k)}}{f} | + ∑_{j=1}^{k−1} |A_{j} || \frac{f^{(j)}}{f} | + | \frac{A_{k}}{f} |.

By Lemma 2.11, we know that f is transcendental with ρ(f) ⩾ σ. By the hypothesis λ(1/f) < σ and Hadamard factorization theorem, we can write f as f(z) = \frac{g(z)}{d(z)} , where g(z) and d(z) are entire functions with λ(d) = ρ(d) = λ(1/f) < σ, ρ(f) = ρ(g) ⩾ σ. By Lemma 2.9, for any given ε (0 < ε < ρ(f)), there exists a set H_{3} ⊂ [0, +∞) with \overline{dens}H_{3} = 1 such that

(3.9)  M(r, g) ⩾ e^{r^{ρ(g)−ε}}

holds for all z satisfying |z| = r ∈ H_{3}. By (3.9), for all z satisfying |z| ∈ H_{3} at which |g(z)| = M(r, g), we get

(3.10)  |g(z)| ⩾ 1.

Then, by (3.10), we obtain

| \frac{A_{k}(z)}{f(z)} | = | \frac{d(z)A_{k}(z)}{g(z)}| ⩽ |d(z)A_{k}(z)|.

We have ρ(d(z)A_{k}(z)) < σ and set

ρ_{1} = max{ρ(A_{j})(j = 1, 2, …, k), ρ(d)} < σ,

hence by using similar arguments as in the proof of Theorem 1.3, for any given ε (0 < ε < σ − ρ_{1}) there exists a set H_{4} = H ∩ H_{3} ⊂ [0, +∞) with positive upper density such that for all z satisfying |z| = r ∈ H_{4} ∖ ([0, 1] ∪ E_{0} ∪ E_{1}), r → +∞, at which |g(z)| = M(r, g), we have

| \frac{f^{(j)} (z)}{f(z)} | ⩽ r[T (2r, f)]^{2k} (j = 1, 2, …, k), (3.11)

| \frac{A_{k}(z)}{f(z)} | ⩽ e^{r^{ρ_{1} +ε}} , (3.12)

|A_{j} (z)| ⩽ e^{r^{ρ_{1} +ε}} (j = 1, 2, …, k − 1), (3.13)

|A_{0}(z)| ⩾ e^{αr^{σ}} . (3.14)

Substituting (3.11)–(3.14) into (3.8), we obtain for all z satisfying |z| = r ∈ H_{4} ∖ ([0, 1] ∪ E_{0} ∪ E_{1}), r → +∞, at which |g(z)| = M(r, g), that

e^{αr^{σ}} ⩽ r[T (2r, f)]^{2k} + ∑_{j=1}^{k−1} e^{r^{ρ_{1} +ε}} r[T (2r, f)]^{2k} + e^{r^{ρ_{1} +ε}} (3.15)

⩽ (k + 1)r[T (2r, f)]^{2k}e^{r^{ρ_{1} +ε}} .

Hence by (3.15), we have ρ(f) = ∞. This is a contradiction which means that the assumption of ρ(f) < ∞ is not true. Hence, we conclude that ρ(f) = ∞.

Since A_{k} ≢ 0, then by Lemma 2.10, we obtain \overline{λ}(f) = λ(f) = ρ(f) = ∞. By (1.3), it is easy to see that if f has a zero at z_{0} of order m, m > k, and A_{j} (j = 0, 1, …, k − 1) are analytic at z_{0}, then A_{k} must have a zero at z_{0} of order m − k. Therefore, we get by A_{k} ≢ 0 that

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

On the other hand, (1.3) may be rewritten as

\frac{1}{f} = \frac{1}{A_{k}} [ \frac{f^{(k)}}{f} + ∑_{j=1}^{k−1} A_{j} \frac{f^{(j)}}{f} + A_{0}].

So, we get

(3.17)  m( r, \frac{1}{f}) ⩽ m( r, \frac{1}{A_{k}} ) + ∑_{j=0}^{k−1} m(r, A_{j}) + ∑_{j=1}^{k} m( r, \frac{f^{(j)}}{f} ) + O(1).

By (3.16) and (3.17), we have

(3.18)  T ( r, \frac{1}{f}) ⩽ k\overline{N}( r, \frac{1}{f}) + N( r, \frac{1}{A_{k}} ) + ∑_{j=0}^{k−1} N(r, A_{j})

+ m( r, \frac{1}{A_{k}} ) + ∑_{j=0}^{k−1} m(r, A_{j}) + ∑_{j=1}^{k} m( r, \frac{f^{(j)}}{f} ) + O(1).

By (3.18) and the lemma of logarithmic derivative [10], there exists a set E of r of a finite linear measure such that for all r ∉ E, we have

(3.19) T (r, f) = T ( r, \frac{1}{f}) + O(1) ⩽ k\overline{N}( r, \frac{1}{f}) + ∑_{j=0}^{k} T (r, A_{j}) + C log(rT (r, f)),

where C is a positive constant. For sufficiently large r, we have

C log(rT (r, f)) ⩽ \frac{1}{2} T (r, f). (3.20)

T (r, A_{j}) ⩽ r^{α+ε} (j = 0, 1, …, k), (3.21)

where α = max{ρ(A_{j}) (j = 0, 1, …, k)}. Then, for r ∉ E sufficiently large, by using (3.20) and (3.21), we conclude from (3.19) that for r ∉ E and r sufficiently large

T (r, f) ⩽ k\overline{N}( r, \frac{1}{f}) + (k + 1)r^{α+ε} + \frac{1}{2} T (r, f).

So, we get

(3.22)  T (r, f) ⩽ 2k\overline{N}( r, \frac{1}{f}) + 2(k + 1)r^{α+ε} , r ∉ E.

Hence, by (3.22), we get ρ_{2}(f) ⩽ \overline{λ}_{2}(f) and then λ_{2}(f) ⩾ \overline{λ}_{2}(f) ⩾ ρ_{2}(f). Since by definition, we have \overline{λ}_{2}(f) ⩽ λ_{2}(f) ⩽ ρ_{2}(f), therefore

(3.23)  \overline{λ}_{2}(f) = λ_{2}(f) = ρ_{2}(f).

Furthermore, if λ(1/f) < min{μ(f), σ}, then f is a meromorphic solution of (1.3) with ρ(f) = ∞, λ(1/f) < μ(f) and by Remark 1.2, we have max{ρ(A_{j}) : j = 0, 1, …, k} = ρ(A_{0}) = β < ∞. Thus, by Lemma 2.6, we get

(3.24)  ρ_{2}(f) ⩽ β.

By (3.23) and (3.24), we conclude that \overline{λ}_{2}(f) = λ_{2}(f) = ρ_{2}(f) ⩽ ρ(A_{0}). □

Proof of Corollary 1.1. Since A_{0} is a meromorphic function with finitely many poles and ρ(A_{0}) = σ, then by the Hadamard factorization theorem, we can write A_{0} as A_{0}(z) = \frac{B(z)}{P (z)} , where B(z) is an entire function with ρ(A_{0}) = ρ(B) = σ and P (z) is a polynomial. Hence, by Lemma 2.7, for any given ε (0 < ε < σ), there exists a set H_{1} ⊂ [0, +∞) with \overline{dens}H_{1} ⩾ 1 − 2σ > 0 (by Proposition 1.1 we have m(H_{1}) = ∞) such that |B(z)| ⩾ e^{r^{σ−ε}} holds for all z, |z| = r ∈ H_{1} and r → +∞. Hence

(3.25)  |A_{0}(z)| = |\frac{B(z)}{P (z)}| ⩾ \frac{e^{r^{σ−ε}}}{cr^{m}} ⩾ e^{r^{σ−2ε}} ,

where c > 0 is a constant and m = degP (z) ⩾ 1. Since A_{j} (j = 0, 1, …, k) are meromorphic functions having finitely many poles and since the poles of f can only occur at the poles of A_{j} (j = 0, 1, …, k), then f must have finitely many poles. Therefore, λ(1/f) = 0. Then, for any given ε with 0 < 2ε < σ − ρ, we have (3.25) and

(3.26)  ρ = max{ρ(A_{j}) : j = 1, 2, …, k} < σ − 2ε.

Furthermore, all meromorphic solutions f ≢ 0 of (1.2) or (1.3) satisfy

(3.27)  λ(1/f) < σ − 2ε.

Then, from (3.25), (3.26) and (3.27) by application of Theorem 1.3 for equation (1.2), we find that every meromorphic solution f ≢ 0 of (1.2) satisfies μ(f) = ρ(f) = ∞ and σ − 2ε ⩽ ρ_{2}(f) ⩽ σ. By ε (0 < 2ε < σ − ρ) being arbitrary, we obtain ρ_{2}(f) = σ. Furthermore, we conclude from Theorem 1.4 that every meromorphic solution f of equation (1.3) satisfies \overline{λ}(f) = λ(f) = ρ(f) = ∞ and \overline{λ}_{2}(f) = λ_{2}(f) = ρ_{2}(f) and if λ(1/f) < μ(f), then ρ_{2}(f) ⩽ ρ(A_{0}). □

Proof of Corollary 1.2. Since A_{0} is a meromorphic function with λ(\frac{1}{A_{0}} ) < μ(A_{0}) ⩽ ρ(A_{0}) = σ < \frac{1}{2}, then by Lemma 2.8, for any ε (0 < ε < σ), there exists a set H_{2} ⊂ (1, +∞) that has \overline{logdens}H_{2} > 0 (by Proposition 1.1 we have m(H_{2}) = ∞) such that for all z, |z| = r ∈ H_{2}, we have

(3.28)  |A_{0}(z)| ⩾ e^{(1+o(1))r^{σ−ε}} .

Obviously, the poles of f must occur at the poles of A_{j} (j = 0, 1, …, k), note that all poles of f are of uniformly bounded multiplicity, then by λ(1/A_{0}) < σ and ρ = max\{ρ(A_{j}) : j = 1, 2, …, k\} < σ, we have λ(\frac{1}{f}) < σ. Then, for a given ε with 0 < ε < min\{σ − ρ, σ − λ(\frac{1}{f})\}, we have (3.28) and

(3.29)  ρ = max{ρ(A_{j}) : j = 1, 2, …, k} < σ − ε and λ(1/f) < σ − ε.

Then by (3.28) and (3.29) with application of Theorem 1.3 for equation (1.2), we find that every meromorphic solution f ≢ 0 whose poles are of uniformly bounded multiplicity of equation (1.2) satisfies μ(f) = ρ(f) = ∞ and σ − ε ⩽ ρ_{2}(f) ⩽ σ. Since ε (0 < ε < min{σ − ρ, σ − λ(1/f)}) being arbitrary, we obtain ρ_{2}(f) = σ. Furthermore, with application of Theorem 1.4 for equation (1.3), we find that every meromorphic solution f whose poles are of uniformly bounded multiplicity of equation (1.3) satisfies

\overline{λ}(f) = λ(f) = ρ(f) = ∞, \overline{λ}_{2}(f) = λ_{2}(f) = ρ_{2}(f)

and if λ(1/f) < μ(f), then ρ_{2}(f) ⩽ ρ(A_{0}). □

References

1. B. Belaïdi, Estimation of the hyper-order of entire solutions of complex linear ordinary differential equations whose coefficients are entire functions, Electron. J. Qual. Theory Differ. Equ. 5 (2002), 1–8.

2. , Growth of solutions of certain non-homogeneous linear differential equations with entire coefficients, JIPAM. J. Inequal. Pure Appl. Math. 5(2) (2004), Article 40, 1–9.

3. A. Besicovitch, On integral functions of order < 1, Math. Ann. 97(2) (1927), 677–695.

4. Z. X. Chen, Zeros of meromorphic solutions of higher order linear differential equations, Analysis 14(4) (1994), 425–438.

5. , The zero, pole and order of meromorphic solutions of differential equations with meromorphic coefficients, Kodai Math. J. 19(3) (1996), 341–354.

6. Z. X. Chen, C. C. Yang, Some further results on the zeros and growths of entire solutions of second order linear differential equations, Kodai Math. J. 22(2) (1999), 273–285.

7. Z. X. Chen, On the rate of growth of meromorphic solutions of higher order linear differential equations, Acta Math. Sinica 42(3) (1999), 551–558 (in Chinese).

8. Y. Chen, Estimates of the zeros and growths of meromorphic solutions of homogeneous and non-homogeneous second order linear differential equations, Math. Appl. (Wuhan) 23(1) (2010), 18–26.

9. G. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37(1) (1988), 88–104.

10. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

11. K. H. Kwon, On the growth of entire functions satisfying second order linear differential equations, Bull. Korean Math. Soc.33(3) (1996), 487–496.

12. I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Stud. Math. 15, Walter de Gruyter, Berlin, 1993.

13. C. H. Li, Y. X. Gu, On the complex oscillation of differential equations f′′+ e^{az} f′+ Q(z)f = F (z), Acta Math. Sci. 25A (2) (2005), 192–200.

14. X. Shen, H. Y. Xu, The zeros and growth of solutions of higher order differential equations, Fourth International Conference on Computational and Information Sciences (2012), 605–608.

15. C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Math. Appl. 557, Kluwer, Dordrecht, 2003.

Department of Mathematics (Received 22 03 2013) Laboratory of Pure and Applied Mathematics University of Mostaganem (UMAB) B. P. 227 Mostaganem Algeria

 maamaths73@yahoo.fr
 belaidi@univ-mosta.dz