Complex Oscillation of Higher-Order Linear Differential Equations with Coefficients Being Lacunary Series of Finite Iterated Order

Definition 1. The p-iterated order of a meromorphic function f(z) is defined by

Remark 2. If f(z) is an entire function, then the p-iterated order of f(z) is defined by

If p = 1, the classical growth of order of f(z) is defined by If p = 2, the hyperorder of f(z) is defined by

Definition 3. If f(z) is an entire function with 0 < σ_p(f) = σ < ∞, then the p-iterated type of f(z) is defined by

Definition 4. The p-iterated lower order of an entire function f(z) is defined by

Definition 5. The finiteness degree of the iterated order of a meromorphic function f(z) is defined by

Definition 6. The p-iterated exponent of convergence of a-point of a meromorphic function f(z) is defined by

If a = 0, the p-iterated exponent of convergence of zero-sequence and distinct zero-sequence of a meromorphic function f(z) are defined, respectively, by The p-iterated lower exponent of convergence of zero-sequence and distinct zero-sequence of a meromorphic function f(z) are defined, respectively, by

Theorem A (see [8].)Let A₀(z), …, A_k−1(z) be entire functions, if i(A_j) ≤ p  (j = 0, …, k − 1), then i(f) ≤ p + 1 and σ_p+1(f) ≤ max {σ_p(A_j), j = 0, …, k − 1} hold for all solutions of (12).

Theorem B (see [8].)Let A₀(z), …, A_k−1(z) be entire functions and let i(A₀) = p. If i(A_j) < p or σ_p(A_j) < σ_p(A₀) for all j = 1, …, k − 1, then i(f) = p + 1 and σ_p+1(f) = σ_p(A₀) hold for all nontrivial solutions of (12).

Theorem C (see [4], [12].)Let A₀(z), …, A_k−1(z) be entire functions and let i(A₀) = p. Assume that max {σ_p(A_j) : j = 1, …, k − 1} ≤ σ_p(A₀)  (>0) and max {τ_p(A_j) : σ_p(A_j) = σ_p(A₀)} < τ_p(A₀). Then, every solution f(z)≢0 of (12) satisfies i(f) = p + 1 and σ_p+1(f) = σ_p(A₀).

Theorem D (see [10].)Let A₀(z), …, A_k−1(z) be entire functions of finite iterated order satisfying i(A₀) = p, σ_p(A₀) = σ, and $$. Then, every nontrivial solution f(z) of (12) satisfies σ_p+1(f) = σ_p(A₀) = σ.

Theorem E (see [10].)Let A₀(z), …, A_k−1(z) be entire functions of finite iterated order satisfying max {σ_p(A_j), j ≠ 0} < μ_p(A₀) = σ_p(A₀) and $$, where E is a set of r of finite linear measure, then every nontrivial solution f(z) of (12) satisfies σ_p+1(f) = μ_p(A₀) = σ_p(A₀).

Theorem F (see [5].)Let A_j(z)  (j = 0, …, k − 1) be entire functions of finite iterated order such that there exists one transcendental A_d  (0 ≤ d ≤ k − 1) satisfying σ_p(A_j) ≤ σ_p(A_d) < ∞ for all j ≠ s, then (12) has at least one solution f(z) that satisfies i(f) = p + 1 and σ_p+1(f) = σ_p(A_d).

Remark 7. Theorems B–E are investigating the growth of solutions of (12) when the coefficients are of finite iterated order and A₀(z) grows faster than other coefficients in (12). What can we have if there exists one middle coefficient A_d(z)  (1 ≤ d ≤ k − 1) such that A_d(z) grows faster than other coefficients in (12) or (13)? Many authors have investigated this question when A_d(z) is of finite order and obtained many results (e.g., see [13–15]). Here, our question is that under what conditions can we obtain similar results with Theorems B-C if A_d(z)  (1 ≤ d ≤ k − 1) is of finite iterated order and grows faster than other coefficients in (12) or (13).

In 2009, Tu and Liu make use of the proposition of lacunary power series to investigate the above question in the case p = 1 and obtain the following result.

Theorem G (see [15].)Let A_j(z)  (j = 0, …, k − 1),   F(z) be entire functions satisfying max {σ(A_j), j ≠ d,   σ(F)} < σ(A_d) < ∞  (1 ≤ d ≤ k − 1). Suppose that $$ is an entire function of regular growth such that the sequence of exponents {λ_n} satisfies Fabry gap condition

then one has

• (i)

  if F(z)≡ 0, then every transcendental solution f(z) of (13) satisfies σ₂(f) = σ(A_d);

• (ii)

  if F(z)≢0, then every transcendental solution f(z) of (13) satisfies $$.

Theorem 8. Let A_j(z)  (j = 0, …, k − 1),   F(z) be entire functions of finite iterated order and satisfying 0 < max {σ_p(A_j), j ≠ d} ≤ σ_p(A_d) < ∞ and max {τ_p(A_j) : σ_p(A_j) = σ_p(A_d)} < τ_p(A_d)  (0 ≤ d ≤ k − 1). Suppose that $$ is an entire function such that the sequence of exponents {λ_n} satisfies

for some η > 0, then one has

• (i)

  if σ_p(F) < σ_p(A_d) or σ_p(F) = σ_p(A_d) and τ_p(F) < τ_p(A_d), then every transcendental solution f(z) of (13) satisfies σ_p+1(f) = σ_p(A_d); furthermore if F(z)≢0, then every transcendental solution f(z) of (13) satisfies $$;

• (ii)

  if σ_p(F) > σ_p(A_d) and σ_p+1(F) ≤ σ_p(A_d), then all solutions of (13) satisfy σ_p(f) ≥ σ_p(F) and σ_p+1(f) ≤ σ_p(A_d);

• (iii)

  if σ_p+1(F) > σ_p(A_d), then all solutions of (13) satisfy σ_p+1(f) = σ_p+1(F), and $$ holds for all solutions of (13) with at most one exceptional solution f₀ satisfying λ_p+1(f₀) < σ_p+1(F).

Remark 9. If $$ is an entire function of finite order and the sequence of exponents {λ_n} satisfies (14), then (18) in Lemma 15 holds for f(z), but for entire functions of infinite order, (14) certainly does not imply (18) in Lemma 15 (see [8]); therefore, we need more stringent gap condition (15) which is sufficient and unnecessary for Theorem 8.

Theorem 10. Let A_j(z)  (j = 0, …, k − 1),   F(z) be entire functions of finite iterated order satisfying max {σ_p(A_j),j ≠ d, σ_p(F)} < μ_p(A_d) = σ_p(A_d) = σ < ∞  (0 ≤ d ≤ k − 1). Suppose that $$ is an entire function such that the sequence of exponents {λ_n} satisfies gap condition (15), then every transcendental solution f(z) of (13) satisfies μ_p+1(f) = σ_p+1(f) = σ. Furthermore if F(z)≢0, then every transcendental solution f(z) of (13) satisfies $$.

Theorem 11. Let A_j(z)  (j = 0, …, k − 1),   F(z) be entire functions satisfying max {σ_p(A_j),j ≠ d} < σ_p(A_d) < ∞  (0 ≤ d ≤ k − 1). Suppose that T(r, A_d) ~ log M(r, A_d) as r → ∞ outside a set of r of finite logarithmic measure, then one has

• (i)

  if σ_p(F) < σ_p(A_d), then every transcendental solution f(z) of (13) satisfies σ_p+1(f) = σ_p(A_d); furthermore, if F(z)≢0, then every transcendental solution f(z) of (13) satisfies $$;

• (ii)

  if σ_p(F) > σ_p(A_d) and σ_p+1(F) ≤ σ_p(A_d), then all solutions of (13) satisfy σ_p(f) ≥ σ_p(F) and σ_p+1(f) ≤ σ_p(A_d);

• (iii)

  if σ_p+1(F) > σ_p(A_d), then all solutions of (13) satisfy σ_p+1(f) = σ_p+1(F), and $$ holds for all solutions of (13) with at most one exceptional solution f₀ satisfying λ_p+1(f₀) < σ_p+1(F);

• (iv)

  if μ_p(A_d) = σ_p(A_d) = σ and F(z)≢0 and σ_p(F) < σ, then every transcendental solution f(z) of (13) satisfies $$.

Remark 12. Theorem 10 implies that all the solutions of (13) are of regular growth if A_d is of regular growth under some conditions; Theorem 11 is an improvement of the Theorem in [14, page 2694] and Theorems 1-2 in [16, page 624]. In fact, by Lemma 15, the gap condition (15) in Theorem 8 implies that T(r, A_d) ~ log M(r, A_d) as r → ∞ outside a set of r of finite logarithmic measure; therefore, Theorem 11 is a generalization of Theorem 8 in a sense, but the condition on A_d in Theorem 8 is more stringent than that in Theorem 11. In addition, Theorems 8–11 may have polynomial solutions of degree <d if d > 1.

Lemma 13 (see [17].)Let f(z) be a transcendental meromorphic function, and let α > 1 be a given constant, for any given constant and for any given ε > 0,

• (i)

  there exist a constant B > 0 and a set E₁ ⊂ (0, +∞) having finite logarithmic measure such that for all z satisfying |z | = r ∉ E₁, one has

  $$ ()

• (ii)

  There exists a set H₁ ⊂ [0,2π) that has linear measure zero a constant B > 0 that depends only on α, for any θ ∈ [0,2π)∖H₁, there exists a constant R₀ = R₀(θ) > 1 such that for all z satisfying argz = θ and |z | = r > R₀, one has

  $$ ()

Remark 14. Throughout this paper, we use E₁ ⊂ (0, ∞) to denote a set having finite logarithmic measure or finite linear measure, not always the same at each occurrence.

Lemma 15 (see [18].)Let $$ be an entire function and the sequence of exponents {λ_n} satisfies the gap condition (15). Then for any given ε > 0,

holds outside a set of finite logarithmic measure, where $$, $$.

Lemma 16 (see [4].)Let f(z) be an entire function of finite iterated order satisfying 0 < σ_p(f) = σ < ∞ and τ_p(f) = τ > 0, then for any given β < τ, there exists a set E₂ ⊂ (0, +∞) having infinite logarithmic measure such that for all r ∈ E₂, one has

Lemma 17. Let $$ be an entire function of finite iterated order satisfying 0 < σ_p(f) = σ < ∞ and τ_p(f) = τ > 0 such that the sequence of exponents {λ_n} satisfies the gap condition (15). Then, for any given β > τ, there exists a set E₃ ⊂ (0, +∞) having infinite logarithmic measure such that for all |z| = r ∈ E₃, one has

Proof. By Lemma 15, for any given ε > 0, we have

For any given β < τ, we can choose β₁ and sufficiently small ε > 0 such that β < β₁ < τ and β < (1 − ε)β₁ < τ; by Lemma 16, there exists a set E₂ ⊂ (0, +∞) having infinite logarithmic measure such that for all |z| = r ∈ E₂∖E₁, we have where E₃ = E₂∖E₁ is a set having infinite logarithmic measure.

Lemma 18 (see [13].)Let f(z) be a transcendental entire function. Then, there is a set E₁ ⊂ (0, +∞) having finite logarithmic measure such that for all z satisfying |z | = r ∉ E₁ and |f(z)| = M(r, f), one has

Lemma 19 (see [7], [9], [10].)Let A₀(z), …, A_k−1(z),   F(z)≢0 be meromorphic functions, and let f(z) be a meromorphic solution of (13) satisfying one of the following conditions:

• (i)

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

• (ii)

  b = max {σ_p+1(F), σ_p+1(A_j)(j = 0, …, k − 1)} < σ_p+1(f),

then $$.

Lemma 20 (see [8].)Let A₀(z), …, A_k−1(z),   F(z) be entire functions of finite iterated order, if i(A_j) ≤ p,   i(F) ≤ p  (j = 0, …, k − 1). Then σ_p+1(f) ≤ max {σ_p(A_j), σ_p(F), j = 0, …, k − 1} holds for all solutions of (13).

Lemma 21 (see [2].)Let g : (0, +∞) → R, h : (0, +∞) → R be monotone increasing functions such that

• (i)

  g(r) ≤ h(r) outside of an exceptional set of finite linear measure. Then, for any α > 1, there exists r₀ > 0 such that g(r) ≤ h(αr) for all r > r₀.

• (ii)

  g(r) ≤ h(r) outside of an exceptional set of finite logarithmic measure. Then, for any α > 1, there exists r₀ > 0 such that g(r) ≤ h(r^α) for all r > r₀.

Lemma 22 (see [19].)Let f(z) be an entire function of finite iterated order satisfying μ_p(f) = μ < ∞. Then, for any given ε > 0, there exists a set E₄ ⊂ (0, +∞) having infinite logarithmic measure such that for all r ∈ E₄, one has

Lemma 23 (see [2], [20].)Let f(z) be a transcendental entire function, let 0 < η₁ < 1/4 and z_r a point such that |z_r | = r and that $$ holds. Then, there exists a set E₁ ⊂ (0, +∞) of finite logarithmic measure such that

holds for all j ∈ ℕ and all r ∉ E₁, where ν_f(r) is the central index of f(z).

Lemma 24 (see [7], [9].)Let f(z) be an entire function of finite iterated order satisfying σ_p(f) = σ,   μ_q(f) = μ,   p, q ∈ ℕ. Then, one has

Lemma 25. Let A_j(z)  (j = 0, …, k − 1),   F(z) be entire functions of finite iterated order satisfying max {σ_p(A_j), j ≠ d, σ_p(F)} ≤ μ_p(A_d) < ∞. Then, every solution f(z) of (13) satisfies μ_p+1(f) ≤ μ_p(A_d).

Proof. By (13), we have

By Lemma 23, there exists a set E₁ having finite logarithmic measure such that for all |z | = r ∉ E₁ and |f(z)| = M(r, f), we have By Lemma 22, for any given ε > 0, there exists a set E₄ having infinite logarithmic measure such that for all |z | = r ∈ E₄ and |f(z)| = M(r, f) > 1, we have Hence from (27)–(29), for any given ε > 0 and for all z satisfying |z | = r ∈ E₄∖E₁ and |f(z)| = M(r, f), we have By (30) and Lemma 24, we have μ_p+1(f) ≤ μ_p(A_d). Therefore, we complete the proof of Lemma 25.

Lemma 26. Let A₀, A₁, …, A_k−1,   F≢0 be meromorphic functions of finite iterated order; if f is a meromorphic solution of the (13) and satisfies b = max {σ_p+1(F), σ_p+1(A_j), j = 0, …, k − 1} < μ_p+1(f), then $$.

Proof. By (13), we have

By (31), we get By the lemma of logarithmic derivative and (31), we have By (32) and (33), we have Since max {σ_p+1(F), σ_p+1(A_j),  j = 0, 1, …, k − 1} < μ_p+1(f), for sufficiently large r, we have

By (34)-(35), we have

By Lemma 21  (i) and by (36), we have $$.

Lemma 27. Let f(z) be a transcendental entire function, for each sufficiently large |z | = r, and let $$ be a point satisfying |f(z_r)| = M(r, f). Then, there exists a constant δ_r  (>0) such that for all z satisfying |z | = r ∉ E₁ and argz = θ ∈ [θ_r − δ_r, θ_r + δ_r], one has

Proof. If $$ is a point satisfying |f(z_r)| = M(r, f), since |f(z)| is continuous in |z | = r, then there exists a constant δ_r  (>0) such that for all z satisfying |z | = r (large enough) and argz = θ ∈ [θ_r − δ_r, θ_r + δ_r], we have

By Lemma 23, we have holds for all z satisfying |z | = r ∉ E₁ and argz = θ ∈ [θ_r − δ_r, θ_r + δ_r].

Lemma 28. Let f(z) be a transcendental entire function, for each sufficiently large |z | = r, and let $$ be a point satisfying |f(z_r)| = M(r, f). Then, there exists a constant δ_r  (>0) such that for all z satisfying |z | = r ∉ E₁ and argz = θ ∈ [θ_r   −   δ_r, θ_r   +   δ_r], one has

Proof. If $$ is a point satisfying |f(z_r)| = M(r, f), then by Lemma 27 there exists a constant δ_r  (>0) such that for all z satisfying |z | = r∉E₁ and argz = θ ∈ [θ_r   −   δ_r, θ_r   +   δ_r], we have

Since f(z) is transcendental, we have ν_f(r) → ∞  (r → ∞). Hence by (41), for all z satisfying |z | = r ∉ E₁ and argz = θ ∈ [θ_r − δ_r, θ_r + δ_r], we have Therefore, we complete the proof of Lemma 28.

Lemma 29. Let f(z) be an entire function of order 0 < σ_p(f) = σ < ∞. Then for any given ε > 0, there exists a set E₅ ⊂ (0, +∞) with positive upper logarithmic density such that for all |z | = r ∈ E₅, one has

Proof. Since 0 < σ_p(f) = σ < ∞, then for any given ε > 0, there exists an increasing sequence {r_n} tending to ∞ such that for n ≥ n₄, we have

Since ε > 0, we can choose α to satisfy 1 < α < (σ − (ε/2))/(σ − ε). Then for all $$, we have Setting $$, we have Thus, Lemma 29 is proved.

Lemma 30. Let f(z) be a transcendental entire function satisfying 0 < σ_p(f) = σ < ∞ and T(r, f) ~ log M(r, f) as r → ∞ outside a set r of finite logarithmic measure. Then for any given ε > 0, there exists a set E₆ ⊂ (0, +∞) with positive upper logarithmic density and a set H₂ ⊂ [0,2π) with linear measure zero such that for all z satisfying r ∈ E₆ and argz = θ ∈ [0,2π)∖H₂, one has

Proof. Since m(r, f) ~ log M(r, f) as r → ∞  (r ∉ E₁), then by the definition of m(r, f), there exists a set H₂ ⊂ [0,2π) having linear measure zero such that for all z satisfying argz = θ ∈ [0,2π)∖H₂, we have

By Lemma 29, for any given ε > 0, there exists a set E₆ ⊂ (0, +∞) with positive upper logarithmic density, we have By (48) and (49), for any given ε > 0 and for all z satisfying |z | = r ∈ E₆∖E₁ and argz = θ ∈ [0,2π)∖H₂, we have Therefore, we complete the proof of Lemma 30.

Proof of Theorem 8. (i) Assume that f(z) is a transcendental solution of (13). By (13), we have

By Lemma 13  (i), there exists a set E₁ ⊂ [1, ∞) having finite logarithmic measure and a constant B > 0 such that holds for all |z | = r ∉ E₁ and for sufficiently large r. Since max {σ_p(A_j), j ≠ d, σ_p(F)} ≤ σ_p(A_d) and max {τ_p(A_j), τ_p(F) : σ_p(A_j) = σ_p(A_d) = σ_p(F)} < τ_p(A_d), we choose α₁, β₁ to satisfy max {τ_p(A_j), j ≠ d, τ_p(F)} < α₁ < β₁ < τ_p(A_d); by Lemma 17, there exists a set E₃ ⊂ (0, +∞) having infinite logarithmic measure such that for all z satisfying |z | = r ∈ E₃ and for sufficiently large r, we have By Lemma 18, there exists a set E₁ ⊂ (0, +∞) having finite logarithmic measure such that for all z satisfying |f(z)| = M(r, f) > 1 and |z | = r ∉ E₁, we have Hence from (51)–(54), for all z satisfying |z | = r ∈ E₃∖E₁ and |f(z)| = M(r, f), we have By (55), we have

On the other hand, by Lemma 20, we have σ_p+1(f) ≤ σ_p(A_d). Therefore, every transcendental solution f(z) of (13) satisfies σ_p+1(f) = σ_p(A_d). Furthermore if F(z)≢0, then by Lemma 19, we have that every transcendental solution f(z) of (13) satisfies $$.

(ii) We assume that f is a solution of (13). By the elementary theory of differential equations, all the solutions of (13) are entire functions and have the form

where C₁, …, C_k are complex constants, {f₁, …, f_k} is a solution base of (12), and f^* is a solution of (13) and has the form where D₁, …, D_k are certain entire functions satisfying where G_j(f₁, …, f_k) are differential polynomials in f₁, …, f_k and their derivative with constant coefficients, and W(f₁, …, f_k) is the Wronskian of f₁, …, f_k. By Theorem A, we have σ_p+1(f_j) ≤ σ_p(A_d)  (j = 1, 2, …, k); then by (57)–(59), we get

Since σ_p(F) > σ_p(A_d), it is easy to see that σ_p(f) ≥ σ_p(F) by (13).

(iii) Suppose that f is a solution of (13), it is easy to see that σ_p+1(f) ≥ σ_p+1(F) by (13). On the other hand, since σ_p+1(F) > σ_p(A_d) and by (57)–(59), we have

Therefore, all solutions of (13) satisfy σ_p+1(f) = σ_p+1(F).

By the same proof in Theorem 4.2 in [8, page 401], we can obtain that all solutions of (13) satisfying $$ with at most one exceptional solution f₀ satisfying λ_p+1(f₀) < σ_p+1(F).

Proof of Theorem 10. Suppose that f(z) is a transcendental solution of (13), by the same proof in Theorem 8, we have σ_p+1(f) = σ_p(A_d) = σ. Thus, it remains to show that μ_p+1(f) = μ_p(A_d) = σ. We choose α₂, β₂ to satisfy

Since the sequence of exponents {λ_n} of A_d satisfies (15) and μ_p(A_d) = σ, then by Lemma 15, there exists a set E₁ having finite logarithmic measure such that for all sufficiently large r ∉ E₁, we have Hence from (51), (52), (54), and (63), for all z satisfying |z | = r ∉ E₁ and |f(z)| = M(r, f), we have Since β₂ is arbitrarily close to σ, by (64) and Lemma 21  (ii), we have On the other hand, by Lemma 26, we have μ_p+1(f) ≤ μ_p(A_d) = σ; therefore, every transcendental solution of (13) satisfies μ_p+1(f) = σ.

Proof of Theorem 11. (i) By Lemma 20, we know that every solution of (13) satisfies σ_p+1(f) ≤ σ_p(A_d). In the following, we show that every transcendental solution f(z) of (13) satisfies σ_p+1(f) ≥ σ_p(A_d). Suppose that f(z) is a transcendental solution of (13). For each sufficiently large circle |z | = r, we take a point $$ satisfying |f(z_r)| = M(r, f). By Lemma 28, there exist a constant δ_r > 0 and a set E₁ such that for all z satisfying |z | = r ∉ E₁ and argz = θ ∈ [θ_r   −   δ_r, θ_r   +   δ_r], we have

By Lemma 13  (ii), there exist a set H₁ ⊂ [0,2π) having linear measure zero and a constant B > 0 such that for sufficiently large r and for all z satisfying argz = θ ∈ [θ_r   −  δ_r, θ_r  +   δ_r]∖H₁, we have Setting max {σ_p(A_j), σ_p(F), j ≠ d} = b < σ_p(A_d), for all z satisfying |z | = r ∉ E₁ and argz = θ ∈ [θ_r − δ_r, θ_r + δ_r] and for any given ε  (0 < 2ε < σ_p(A_d) − b), we have Since T(r, A_d) ~ log M(r, A_d) as r → ∞  (r ∉ E₁), by Lemma 30, for any given ε > 0, there exists a set E₆ ⊂ (0, ∞) with $$ and a set H₂ ⊂ [0,2π) with linear measure zero such that for all z satisfying |z | = r ∈ E₆ and argz = θ ∈ [θ_r − δ_r, θ_r + δ_r]∖H₂, we have Substituting (66)–(69) into (51), for all z satisfying |z | = r ∈ E₆∖E₁ and argz = θ ∈ [θ_r − δ_r, θ_r + δ_r]∖(H₁ ∪ H₂), we have

From (70), we have σ_p+1(f) ≥ σ_p(A_d). Therefore, every transcendental solution f(z) of (13) satisfies σ_p+1(f) = σ_p(A_d). Furthermore, if F(z)≢0, then every transcendental solution f(z) of (13) satisfies $$.

(ii)–(iv) By the same proof in Theorems 8 and 10, we can obtain the conclusions (ii)–(iv).