In this paper, we shall assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions [6, 12]. In what follows, we give the necessary notations and basic definitions.

Definition 1.1([6, 12])

Let f be a meromorphic function. Then the order σ (f) of f (z) is defined by

where T (r, f) is the Nevanlinna characteristic function of f. If f is an entire function, then the order σ (f) of f (z) is defined by

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

Definition 1.2([6, 12])

Let f be a meromorphic function. Then the exponent of convergence of the sequence of zeros of f (z) is defined by

where N (r,1f) is the integrated counting function of zeros of f (z) in {z : |z| ≤ r}. Similarly, the exponent of convergence of the sequence of distinct zeros of f (z) is defined by

where N¯ (r,1f) is the integrated counting function of distinct zeros of f (z) in {z : |z| ≤ r}.

In [10], Wang and Laine investigated the growth of solutions of some second order nonhomogeneous linear differential equation and obtained.

Theorem A([10])

Let A_j (z) (≢ 0) (j = 0, 1) and F (z) be entire functions with max{σ (A_j) (j = 0, 1), σ (F)} < 1, and let a, b be complex constants that satisfy ab ≠ 0 and a ≠ b. Then every nontrivial solution f of the differential equation

is of infinite order.

In this paper, we offer a higher-order result related to Theorem A. In fact we will prove the following results.

Theorem 1.1

Let A_j (z) (≢ 0) (j = 0, 1, · · ·, k − 1), F_j (z) (≢ 0) (j = 1, 2) be entire functions with σ (A_j) < 1 (j = 0, 1, · · ·, k − 1) and σ (F_j) < 1 (j = 1, 2), a and b be non-zero complex numbers such that b = ca (0 < c < 1). Suppose the following:

• there is exactly one s (0 ≤ s ≤ k − 1) such thatσ (As)>max {σ (Aj):j=0,1,⋯,k-1 and j≠s},

• for any τ satisfying 0 < τ < σ(A_s), there exists a subset H ⊂ (1,+∞) with infinite logarithmic measure, such that when |z| = r ∈ H,log∣As (z)∣ >rτ.

Then every solution f of the differential equation

has infinite order.

Corollary 1.1

Let A_j (z) (≢ 0) (j = 0, 1, · · ·, k − 1), F_j (z) (≢ 0) (j = 1, 2) be entire functions with σ (Aj)<12 (j=0,1,⋯,k-1) and σ (F_j) < 1 (j = 1, 2), a and b be non-zero complex numbers such that b = ca (0 < c < 1). Suppose that there is exactly one s (0 ≤ s ≤ k − 1) such that

Then every solution f of the differential equation (1.1) has infinite order.

Remark 1.1

By the hypothesis of Corollary 1.1, we see that 0<σ (As)<12.

Theorem 1.2

Under the hypotheses of Theorem 1.1, suppose further that ϕ (z) ≢ 0 is an entire function with finite order. Then every solution f of (1.1) satisfies

Lemma 2.1([4, 9])

Let P₁, P₂, · · ·, P_n( n ≥ 1) be non-constant polynomials with degree d₁, d₂, · · ·, d_n, respectively, such that deg (P_i − P_j) = max{d_i, d_j} for i ≠ j. Let A (z)-∑j=1nBj (z) ePj(z) where B_j (z) (≢0) are entire functions with σ (B_j) < d_j. Then σ (A)=max1≤j≤n {dj}.

Lemma 2.2([8])

Suppose that P (z) = (α + iβ) zⁿ +· · · ( α, β are real numbers, |α|+|β| ≠ 0) is a polynomial with degree n ≥ 1, that A(z) (≢ 0) is an entire function with σ (A) < n. Set g (z) = A(z) e^P(z), z = re^iθ, δ (P, θ) = α cos nθ − β sin nθ. Then for any given ɛ > 0, there is a set E₁ ⊂ [0, 2π) that has linear measure zero, such that for any θ ∈ [0, 2π)(E₁ ∪ E₂), there is R > 0, such that for |z| = r > R, we have

• if δ (P, θ) > 0, thenexp {(1-ɛ) δ (P,θ) rn}≤ ∣g (reiθ)∣ ≤exp {(1+ɛ) δ (P,θ) rn},

• if δ (P, θ) < 0, thenexp {(1+ɛ) δ (P,θ) rn}≤ ∣g (reiθ)∣ ≤exp {(1-ɛ) δ (P,θ) rn},

where E₂ = {θ ∈ [0, 2π) : δ (P, θ) = 0} is a finite set.

Lemma 2.3([5])

Let f be a transcendental meromorphic function of finite order σ. Let ɛ > 0 be a constant, k and j be integers satisfying k > j ≥ 0. Then the following two statements hold:

• There exists a set E₃ ⊂ (1,+∞) which has finite logarithmic measure, such that for all z satisfying |z| / ∈ E₃ ∪ [0, 1], we have(2.1)|f(k) (z)f(j) (z)|≤ ∣z∣(k-j)(σ-1+ɛ).

• There exists a set E₄ ⊂ [0, 2π) which has linear measure zero, such that if θ ∈ [0, 2π)E₄, then there is a constant R = R(θ) > 0 such that (2.1) holds for all z satisfying arg z = θ and |z| ≥ R.

Lemma 2.4([11])

Let f (z) be an entire function and suppose that

is unbounded on some ray arg z = θ with constant ρ > 0. Then there exists an infinite sequence of points z_n = r_ne^iθ( n = 1, 2, · · ·), where r_n → +∞, such that G(z_n)→∞ and

as n → +∞.

Lemma 2.5([11])

Let f (z) be an entire function with σ (f) = σ < +∞. Suppose that there exists a set E₅ ⊂ [0, 2π) which has linear measure zero, such that log⁺ |f (re^iθ)| ≤ Mr^ρfor any ray arg z = θ ∈ [0, 2π)E₅, where M is a positive constant depending on θ, while ρ is a positive constant independent of θ. Then σ (f) ≤ ρ.

Lemma 2.6([3])

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 differential equation

then f satisfies

Remark 2.1([1, 2])

Let h (z) be a transcendental entire function with order σ (h)=σ<12. Then there exists a subset H ⊂ (1,+∞) having infinite logarithmic measure, such that if σ = 0, then

if σ > 0, then for any α (0 < α < σ),

Proof of Theorem 1.1

We know that b = ca (0 < c < 1), then by (1.1) we get

First, we prove that every solution f of (1.1) satisfies σ (f) ≥ 1. We assume that σ (f) < 1. Obviously σ (A_jf^(j)) < 1 (j = 0, 1, · · ·, k − 1). Rewrite (3.1) as

By (3.2) and the Lemma 2.1, we have

This is a contradiction. Hence, σ (f) ≥ 1. Therefore f is a transcendental solution of equation (1.1).

Now, we prove that σ (f) = +∞. Suppose that σ (f) = σ < +∞. Set

It is clear that, 0 ≤ β < α < 1 and 0 ≤ γ < 1. Then for any given ɛ with 0 < ɛ < min {1 − α, 1 − β, 1 − γ} and for sufficiently large r, we have

By Lemma 2.2, there exists a set E ⊂ [0, 2π) of linear measure zero, such that whenever θ ∈ [0, 2π)E, then δ (az, θ) ≠ 0. By Lemma 2.3, there exists a set E₄ ⊂ [0, 2π) which has linear measure zero, such that if θ ∈ [0, 2π)E₄, then there is a constant R = R(θ) > 1 such that for all z satisfying arg z = θ and |z| ≥ R, we have

By the hypothesis of Theorem 1.1, we see that there exists a subset H ⊂ (1,+∞) having infinite logarithmic measure and τ > 0 such that 0 ≤ β < τ < σ(A_s),

For any fixed θ ∈ [0, 2π)(E ∪ E₄), set

We can obtain

then δ₁ ≠ 0, δ₂ ≠ 0. We now discuss two cases separately.

Proof of Corollary 1.1

By the hypothesis of Corollary 1.1, we see that 0<σ (As)<12. Using Remark 2.1 and using the same reasoning as above, we can get σ (f) = +∞.

Proof of Theorem 1.2

Suppose that f is a solution of equation (1.1). Then, by Theorem 1.1 we have σ (f) = +∞. Set g (z) = f (z) − ϕ (z), g (z) is an entire function and σ (g) = σ (f) = +∞. Substituting f = g + ϕ into (1.1), we obtain

where

We prove that D ≢ 0. In fact, if D ≡ 0, then

Hence σ (ϕ) = +∞, which is a contradiction. Therefore D ≢ 0. We know that the functions A_j (j = 0, 1, · · ·, k − 1), D are of finite order. By Lemma 2.6 and (3.30) we have

Therefore

which completes the proof.

The authors are grateful to the referee and the editor for their valuable comments which lead to the improvement of this paper.