Value distributions of solutions to complex linear differential equations in angular domains

Definition 1.1

The iterated n-order ρ_n,Δ(f) of a meromorphic function f(z) in Δ is defined by

where log[1]⁡r=log⁡randlog[n+1]⁡r=log⁡(log[n]⁡r),n∈N.

Definition 1.2

The growth index of the iterated order of a meromorphic function f(z) in Δ is defined by

Definition 1.3

For n ∈ ℕ and a ∈ ℂ∪ {∞}, the iterated n-convergent exponent of the sequence of a-point in Δ of a meromorphic function f in Δ is defined by

and λ¯n,△(f−a) , the iterated n-convergent exponent of the sequence of distinct a-point in Δ of a meromorphic function f in Δ is defined by

The growth and oscillation of solutions to higher-order linear differential equations in ℂ and in Δ have been well studied by many authors. In the paper [6], Cao and Yi studied the properties of solutions to the arbitrary order linear differential equations in Δ of the form

where A₀(≢ 0), A₁, …, A_k are analytic in Δ. In fact, they got the following theorem.

Theorem 1.4

Let 0 < p < ∞ and i(A₀) = p. If max{i(A_j) : j = 1, 2, …, k} < p or max{ρ_{p, Δ}(A_j) : j = 1, 2, …, k} < ρ_{p, Δ}(A₀), then i(f) = p + 1 and ρ_{p, Δ}(A₀) ≤ ρ_p + 1, Δ(f) holds for all solutions f ≢ 0 of equation (2).

In what follows, we give some notations and definitions of a meromorphic function in an angular domain Ω(α, β) = {z : α < arg z < β}. In this paper, Ω usually denotes the angular domain Ω(α, β) and Ω_ε = {z : α + ε < arg z < β − ε}, where 0<ϵ<β−α2. Let f(z) be a meromorphic function on Ω¯(α,β)={z:α≤arg⁡z≤β}. Recall the definition of Ahlfors-Shimizu characteristic in an angular domain; see [5, pp.66]. Set Ω(r) = Ω(α, β) ∩ {z : 0 < |z| < r}. Define

The order and lower order of f on Ω are defined by

We remark that the above definitions is reasonable because 𝓣(r, ℂ, f) = T(r, f) + O(1); see [1, pp.20].

Definition 1.5

The iterated n-order ρ_n,Ω(f) of a meromorphic function f(z) in an angular region Ω is defined by

where log[1]⁡r=log⁡randlog[n+1]⁡r=log⁡(log[n]⁡r),n∈N.

Remark 1

It is obvious that ρ_1,Ω(f) = ρ_Ω(f).

Motivated by the definition of a convergent exponent of a-value points of f in Ω in [5, p. 93], we give the following definition.

Definition 1.6

For n ∈ ℕ and a ∈ ℂ∪ {∞}, the iterated n-convergent exponent of the sequence of a-point in Ω of a meromorphic function f in Ω is defined by

and λ¯n,Ω(f−a), the iterated n-convergent exponent of the sequence of distinct a-point in Ω of a meromorphic function f in Ω is defined by

The first purpose of this paper is to study the iterated growth order of solutions to complex linear differential equations in an angular domain. In fact, we obtain the results as follows:

Theorem 1.7

Let A(z) be analytic in angular region Ω(α, β) = {z : α < arg z < β}(0 < β − α < 2π) satisfying either

or

where Ωϵ={z:α+ϵ<arg⁡z<β−ϵ},0<ϵ<β−α2,ω=π/(β−α). Then, all solutions f ≢ 0 of the equation f^(k) + A(z)f = 0 have the order ρ_{n, Ω}(f) = + ∞.

Theorem 1.8

Let A_j(z)(j = 0, 1, …, k − 1) be analytic in an angular region Ω(α, β) = {z : α < arg z < β}(0 < β − α < 2π). If for any small ϵ∈(0,β−α2),ρ1,Ω(Aj)<ρ1,Ωϵ(A0)−ωandρn,Ω(Aj)<ρn,Ωϵ(A0)(n≥2,j= 1, 2, …, k − 1), then all solutions f ≢ 0 of equation

have the order ρ_{n + 1, Ω}(f) ≥ ρ_{n, Ωε}(A₀). In particular, ρ_{n, Ω}(f) = + ∞ if ρ_{n, Ωε}(A₀) > 0.

Theorem 1.9

Let A_j(z)(j = 0, 1, …, k − 1) and g(z) be analytic in an angular region Ω(α, β) = {z : α < arg z < β}(0 < β − α < 2π). Suppose that f ≢ 0 is a solution of equation

such that, for n ≥ 2, max{ρ_n,Ω(A_j), ρ_n,Ω(g)} < ρ_{n, Ωε}(f) and, for n = 1, max{ρ_1,Ω(A_j), ρ_1,Ω(g)} < ρ_1,Ωε(f) − ω. Then ρn,Ωϵ(f)≤λ¯n,Ω(f)=λn,Ω(f)andλ¯n,Ωϵ(f)=λn,Ωϵ(f)≤ρn,Ω(f) for any positive integer n.

Recalling the Nevanlinna theory in an angular domain and following the terms in [1], we set

where ω = π/(β − α), and b_n = |b_n|e^iβ_n are poles of f(z) in Ω(α, β) appearing according to their multiplicities. The Nevanlinna angular characteristic is defined as

Thus, the order and lower order of f on Ω can also be defined by

For a ∈ ℂ ∪ {∞}, the convergence exponent of the sequence of a-point in Ω(α, β) of a meromorphic function f is defined by

According to the inequality, see [5, Theorem 2.4.7],

if ρ_Ω(f) < ∞, then ρ_α,β(f) < ∞.

We consider q pairs of real numbers {α_j, β_j} such that

and the angular domains X=⋃j=1q{z:αj≤arg⁡z≤βj}. For a function f meromorphic in the complex plane ℂ, we define the order of f on X as

It is obvious that ραj,βj(f)≤ρx(f)≤∑j=1qραj,βj(f),j=1,2,…,qandρx(f)=+∞ if and only if there exists at least one 1 ≤ j₀ ≤ q such that ρ_αj0,βj0(f) = + ∞.

In [7], Wu considered the growth of solutions to higher order linear homogeneous differential equations in angular domains. The following theorem was obtained.

Theorem 1.10

Let A₀ be a meromorphic function in ℂ with finite lower order μ < ∞ and nonzero order 0 < ρ ≤ ∞ and δ = δ(∞, A₀) > 0. For q pair of real numbers {α_j, β_j} satisfying (8) and

where λ > 0 with μ ≤ λ ≤ ρ. If A_j(z)(j = 1, 2, …, n) are meromorphic functions in ℂ with T(r, A_j) = o(T(r, A₀ every solution f ≢ 0 to the equation

has the order ρX(f)=+∞inX=⋃j=1q{z:αj≤arg⁡z≤βj}.

For the derivatives of the nonzero solutions to the equation in the above theorem, we can get the following result easily.

Theorem 1.11

Let A₀ be a meromorphic function in ℂ with finite lower order μ < ∞ and nonzero order 0 < ρ ≤ ∞ and δ = δ(∞, A₀) > 0. For q pair of real numbers {α_j, β_j} satisfying (8) and

where λ > 0 with μ ≤ λ ≤ ρ. If A_j(z)(j = 1, 2, …, n) are meromorphic functions in ℂ with T(r, A_j) = o(T(r, A₀)), the derivatives of every solution f ≢ 0 to the equation (10) have the order ρx(f^(p))) = +∞ in X=⋃j=1q{z:αj≤arg⁡z≤βj}, where p is a natural number.

The last result relates to the convergence exponent of the sequence of a-point of the solutions of equation (8) in the angular domain X.

Theorem 1.12

Let A₀ be an entire function in ℂ with finite lower order μ < ∞ and nonzero order 0 < ρ ≤ ∞ and δ = δ(∞, A₀) < 0. For q pair of real numbers {α_j, β_j} satisfying (8) and

where λ > 0 with μ ≤ λ ≤ ρ, if A_j(z)(j = 1, 2, …, n) are entire functions in ℂ with T(r, A_j) = o(T(r, A₀)) then every solution f ≢ 0 to the equation (10) satisfies τX(f−a)=+∞inX=⋃j=1q{z:αj≤arg⁡z≤βj} for a ≠ 0.

Lemma 2.1

([8]). The transformation

maps the angular domain X = {z : α < arg z < β}, (0 < β − α < 2π) conformally onto the unit disk {ζ : |ζ| < 1} in the ζ-plane, and maps z = e^iθ₀ to ζ = 0. The image of X_ε = {z : 1≤ |z|≤ r, α + ε ≤ arg z ≤ β − ε}, (0 < ε < β−α2 ) in the ζ-plane is contained in the disk Δ_h := {ζ : |ζ| < h}, where

On the other hand, the inverse image of the disk Δ_h := {ζ : |ζ| < h}, h > 1 in the z-plane is contained in X ∩ {z:|z| ≤ r}, where

The inverse transformation of (13) is

Remark 2

Note that the conformal mapping (13) is univalent, then we get

and

by Lemma 2.1, the notations η, ω here are similar as that in the following Lemma 2.2. Thus, by the Definition 1.3 and 1.6 we conclude that 1ωλn,Ωε(f(z)−a)≤λn,Δ(f(z(ζ))−a)≤1ωλn,Ω(f(z)−a).

Using Lemma 2.1, the following Lemma 2.2 was proved in [9].

Lemma 2.2

Let f(z) be meromorphic in angular region Ω. For any small ε > 0, write ω=πβ−α,η=εβ−α. Then the following inequalities hold:

where z = z(ζ) is the inverse transformation of (13). Consequently,

Remark 3

From (15), (16) and Definition 1.1, 1.5, we obtain, for n ≥ 2,

Lemma 2.3

([9, 10]). Let f(z) be meromorphic in Ω = {z : α < arg z < β}(0 < β − α < 2π) and z = z(ζ) be the inverse transformation of (13). Write F(ζ) = f(z(ζ)), ψ(ζ) = f^(l)(z(ζ)) . Then

where the coefficients α_j are the polynomials (with numerical coefficients) in the variables V(ζ)( = 1z′(ζ)),V′(ζ),V″(ζ),…. Moreover, we have T(r, α_j) = O(log(1 − r)^{− 1}), j = 1, 2, …, l.

Lemma 2.4

([11]). Let A₀, A₁, …, A_{k − 1} and F(≢) be analytic function in Δ and let f(z) be a solution of equation

such that max{ρn,Δ(F),ρn,Δ(Aj),j=0,1,…,k−1}<ρn,Δ(f).Thenλ¯n,Δ(f)=λn,Δ(f)=ρn,Δ(f).

Lemma 2.5

([12]). Let φ(r) be a nondecreasing, continuous function on ℝ⁺, and let

and H = {r ∈ ℝ⁺:|φ(r)| ≥ r^ρ}. Then

Lemma 2.6

([5, Theorem 2.6.5]). Let f(z) be a meromorphic function in Ω¯(α,β) . Then for τ > 1 and a natural number p, we have

where η is such that 0 < 2η < β − α and K is a constant only depending on τ, η, α and β.

It is important and necessary to determine the relations between C_α,β(r, f) and N(r, Ω, f), which will be helpful in characterizing meromorphic functions in an angle in terms of the number of points of some values.

Lemma 2.7

[5] Let f(z) be a meromorphic function on Ω¯(α,β) . Then the following inequalities hold:

and

where N(t)=N(t,Ω,f)=∫1rn(t,Ω,f)tdt,n(t,Ω,f) is the number of poles of f(z) in Ω ∩ {z : 1 < |z| ≤ t} and N0(t)=N(t,Ωδ,f)=∫1tn(t,Ωδ,f)tdtandΩδ=Ω(α+δ,β−δ). The above two inequalities still hold for C¯andN¯ in the place of C and N.

Note that we may replace the integrated counting function N(r,Ω,1f−a) with unintegrated counting function n(r,Ω,1f−a) in the definition of the convergent exponent, because, see [5, pp. 39],

for 1 < dr and

Lemma 2.8

([1]). Suppose that f(z) is a nonconstant meromorphic function in an angular domain Ω(α, β) with 0 < β − α ≤ 2π. Then:

1. ([l, Chapter 1]) for any complex number a ∈ ℂ

   Sα,βr,1f−a=Sα,β(r,f)+O(1), (26)

2. ([l, p.138]) for any r < R,

   Aα,βr,f′f≤KRrω∫1Rlog+⁡T(t,f)t1+ωdt+log+⁡rR−r+log⁡Rr+1 (27)

   and

   Bα,βr,f′f≤4ωrωmr,f′f (28)

   where ω=πβ−α, and K is a positive constant not depending on r and R.

Lemma 2.9

([5, Corollary 2.2.2]). Let f(z) be an analytic function on Ω¯(α,β) with 0 < β − α ≤ 2π. Then we have

where M(r, Ω, f) = max{|f(te^iθ)| : α ≤ θ ≤ β, 1 ≤ t ≤ r} and K is a positive constant.

Let f(z) be a non-constant entire function and M(r, f) the maximum of |f(z)| on the circle |z| = r, that is M(r,f) = max_{|z| = r}|f(z)|. We have the following relations between M(r, f) and T(r, f).

Lemma 2.10

([4, Theorem 1.4]). Suppose f(z) is a non-constant entire function. Then for 0 ≤ r < R < +∞, we have

Proof of Theorem 1.7

Suppose that f ≢ 0 is a solution of f^(k) + A(z)f = 0 in Ω. Then, by Lemma 2.3, F(ζ) = f(z(ζ)) is a solution of the differential equation

in Δ, where α_j(j = 1, 2, …, k) are described in Lemma 2.3 and T(r, α_j) = O(log(1−r)⁻¹), that is i(α_j) = 0 by Definition 1.2, and B(ζ) = A(z(ζ)). If

by (16) and Definition 1.2, we obtain i(B(z(ζ))) ≥ 1. Thus, by Theorem 1.4, we get i(F) = i(B(z(ζ))) + 1 ≥ 2, that is ρ_1,Δ(F) = + ∞.

If

by (16) and Definition 1.2, we obtain i(B(z(ζ))) ≥ n. Thus, by Theorem 1.4, we get i(F) = i(B(z(ζ))) + 1 ≥ n + 1, that is ρ_n,Δ(F) = + ∞. Combining these results with (17), (18) leads to ρ_n,Ω(f) = + ∞. □

Proofof Theorem 1.8

Suppose that f ≢ 0 is a solution of (4). From (19), we have

Set B_k(ζ) = α_k,k, B_j(ζ) = α_j,k + ∑m=jk−1 A_m(z(ζ))α_{j, m}, (j = 1, 2, …, k − 1), B₀(ζ) = A₀(z(ζ)), then F(ζ) = f(z(ζ)) is a solution of the differential equation

in Δ. Since T(r, α_{j, m}) = O(log(1 − r)⁻¹)(1 ≤ j ≤ m ≤ k) by Lemma 2.3, it follows that

If for any small 0 < ε < β−α2 , ρ_{1, Ω}(A_j) < ρ_{1, Ωε}(A₀) − ω, the conclusion holds by [9, Theorem 1.8]. If ρ_{n, Ω}(A_j) < ρ_{n, Ωε}(A₀)(n ≥ 2, j = 1, 2, …, k − 1), it follows from (34) and (18) that

By B₀(ζ) = A₀(z(ζ)) and (18), we get

Combining (35), (36) and ρ_{n, Δ}(B_k(ζ)) = 0, we deduce that ρ_{n, Δ}(B_j(ζ)) < ρ_n,Δ(B₀(ζ)), j = 1, 2, …, k. Thus, by Theorem 1.4, we obtain i(F(ζ)) = n + 1 and ρ_n + 1,Δ(F(ζ)) > ρ_{n, Δ}(B₀(ζ)). It follows from (18), we get ρ_n + 1, Ω(f) ≥ ρ_{n, Ωε}(A₀). □

Proof of Theorem 1.9

Suppose that f ≢ 0 is a solution of (10). Set F(ζ) = f(z(ζ)) and G(ζ) = f(z(ζ)) by (14). By (19) we also have (32) and denote B_j(ζ) as in the proof of Theorem 1.8. Thus, F(ζ) = f(z(ζ)) is a solution of the nonhomogeneous differential equation

in Δ. For n ≥ 2, by similar arguments as in the proof of Theorem 1.8 we get

for j = 1, 2, …, k − 1. Note that

and ρ_{n, Δ}(B_k(ζ)) = 0, we have max{ρ_{n, Δ}(B_j(ζ)), ρ_{n, Δ}(G(ζ))} < ρ_{n, Δ}(F(ζ)) for j = 1, 2, …, k.

Hence, from Lemma 2.4, we have λ_{n, Δ}(F) = λ_{n, Δ}(F) = ρ_{n, Δ}(F). For n = 1, we can easily obtain

and ρ_{1, Δ}(B_k(ζ)) = 0. Thus, max{ρ_{1, Δ}(B_j), ρ_{1, Δ}(G)} < ρ_{1, Δ}(f). Applying Lemma 16 to (37), we deduce λ_{1, Δ}(F) = λ_{1, Δ}(F) = ρ_{1, Δ}(F).

Finally, by Remark 2 and 3, we obtain ρ_{n, Ωε}(f) ≤ λ_{n, Ω}(f) = λ_{n, Ω}(f) and λ_{n, Ωε}(f) = λ_{n, Ωε}(f) ≤ ρ_{n, Ω}(f) for any positive integer n. □

Proof of Theorem 1.11

Suppose that ρ_X(f^(p)) < + ∞, then for any j = 1, 2, …, q, we have ρ_{αj, βj}(f^(p)) < + ∞. By the Definition 6 and Lemma 2.6, we know that ρ_{αj, βj} (f)< + ∞ for j = 1, 2, …, q. Since ρ_{αj, βj} (f) ≤ ρX(f)≤∑j=1qραj,βj(f), we have ρ_X(f) < + ∞. It is a contradiction to Theorem 1.10. □

Proof of Theorem 1.12

Suppose that f(z)≢ 0 is a solution of equation (10) under the hypotheses of Theorem 1.12. Set g(z) = f(z) − a, substitute it into (10) to obtain

and rewrite it as

Appling Wiman-Valiron theory to (42), similarly as in [3, p. 130], we know that

Therefore, for sufficiently large r, we have

Since T(r, g) = T(r, f) + O(1), then we have

By Theorem 1.10, we know that there exist an angular domain Ω₀(α₀, β₀) ⊂ X satisfying ρ_{α0, β0}(g) = ρ_{α0, β0}(f) = + ∞, and then ρ(g) = + ∞. From (2) of Lemma 2.8, for ε∈(0,π2(ρ(A0)+1)) and Ωθ(θ−ε,θ+ε)⊂ Ω₀(α₀, β₀), we can deduce that

and

outside a set of finite linear measure. By using Lemma 1.6 in [4, p.35], it is easy to see that (45) still holds for g_(j), j = 1, 2, …, n in place of g. Similarly as above, (46) and (47) also hold by using g^(j) instead of g. Therefore, according to the definition of D_α,β(r, g), we can deduce, for each s = 1, 2, …, n,

with an exceptional set of finite linear measure. Combining (42) with (48) and from Lemma 2.9 and 2.10, we get