Lower order and Baker wandering domains of solutions to differential equations with coefficients of exponential growth

DSpace https://erepo.uef.fi

Rinnakkaistallenteet Luonnontieteiden ja metsätieteiden tiedekunta

2019

Lower order and Baker wandering domains of solutions to differential equations with coefficients of

exponential growth

Korhonen, Risto

Elsevier BV

Tieteelliset aikakauslehtiartikkelit

© Elsevier Inc.

CC BY-NC-ND https://creativecommons.org/licenses/by-nc-nd/4.0/

http://dx.doi.org/10.1016/j.jmaa.2019.07.007

https://erepo.uef.fi/handle/123456789/7696

Downloaded from University of Eastern Finland's eRepository

Lower order and Baker wandering domains of solutions to differential equations with coefficients of exponential growth

Risto Korhonen, Jun Wang, Zhuan Ye

PII: S0022-247X(19)30568-2

DOI: https://doi.org/10.1016/j.jmaa.2019.07.007 Reference: YJMAA 23317

To appear in: Journal of Mathematical Analysis and Applications Received date: 28 August 2018

Please cite this article in press as: R. Korhonen et al., Lower order and Baker wandering domains of solutions to differential equations with coefficients of exponential growth,J. Math. Anal. Appl.(2019), https://doi.org/10.1016/j.jmaa.2019.07.007

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

SOLUTIONS TO DIFFERENTIAL EQUATIONS WITH COEFFICIENTS OF EXPONENTIAL GROWTH

RISTO KORHONEN, JUN WANG, AND ZHUAN YE

Abstract. We investigate transcendental entire solutions of complex differen- tial equationsf+A(z)f =H(z), where the entire function A(z) has a growth property similar to the exponential functions, andH(z) is an entire function of order less than that ofA. We first prove that the lower order of the entire solu- tion to the equation is infinity. By using our result on the lower order, we prove the entire solution does not bear any Baker wandering domains.

1. Introduction and main results

Let f : C → C be a transcendental meromorphic function, where C is the complex plane and C=C∪ {∞}. We use ρ(f) and μ(f) to denote the order and the lower order off, respectively, which are defined as [20, Definition 1.6]

ρ(f) = lim sup

r→∞

log⁺T(r, f)

logr , μ(f) = lim inf

r→∞

log⁺T(r, f) logr .

Nevanlinna theory is an important tool in this paper, its usual notations and basic results come mainly from [7, 9, 13, 20].

The nth iterate fⁿ of f is defined for all z ∈ C except for a countable set of poles of f, f²,· · · , fⁿ⁻¹. The Fatou set F(f) of f is the set where the family of iterates{fⁿ} is well defined and forms a normal family. The complement of F(f) is called the Julia set J(f) of f. It is well-known that F(f) is open and forward invariant. A basic theory of complex dynamics can be found in, for instance, [5]

for meromorphic functions, and [17] for entire functions.

It is well-known that if U is a Fatou component U then U is connected and fⁿ(U) must be contained in a Fatou component U_n. If all U_n are different, then U is called a wandering domain of f. By Sullivan’s theorem, rational functions have no wandering domains, but wandering domains do occur for transcendental functions. For example, f(z) = z + sinz + 2π has a bounded simply connected wandering domain, see [3] and [17, p. 311]. Baker [1, 2] also gave examples of multiply connected wandering domains, which are now known as Baker wandering domains.

For the convenience of the reader, we give the following definition.

2010 Mathematics Subject Classification. 34M10, 37F10, 30D35.

Key words and phrases. Lower order, Baker wandering domain, entire function, linear differ- ential equation.

1

Definition 1. Let f be a meromorphic function in C. For a wandering domain U, if all U_n are multiply connected components ofF(f) which surround the origin, and the Euclidean distance dist(0, U_n) → +∞ as n → +∞, then U is called a Baker wandering domain.

Clearly, transcendental entire functions without Baker wandering domains have only simply connected Fatou components. There are already some criteria of non- existence of Baker wandering domains [3, 5, 21]. For example, if the entire function f has a path to ∞on whichf is bounded, then all components ofF(f) are simply connected (see [3, Corollary]). On the other hand, if f is entire with a finite deficient value of Nevanlinna, then every component of F(f) is simply connected [21, Corollary 4].

We recall a growth property of complex exponential functionse^p(z), wherep(z) = α_nzⁿ+· · · is a polynomial of degree n, [15, p. 254]. The complex plane is divided into 2n equal open sectors by the rays

argz =−argα_n

n + (2j−1)π

2n (j = 0,1,· · · ,2n).

In each of these sectors, e^p(z) has two possible growth types: it either (1) blows up exponentially; or (2) decays to zero exponentially. In addition,ρ(e^p(z)) =μ(e^p(z)) = n. By [3, Corollary] and the radial behaviour of e^p(z), it is easy to see that e^p(z) has no Baker wandering domains. Inspired by the observation, we want to extend our studies to any transcendental entire functions with nice radial growth or decay properties.

Definition 2. Let A(z) be transcendental entire with ρ(A) = μ(A) < ∞, and δ_A(θ) be a real-valued function defined on [0,2π) which is continuous outside an exceptional set F of finitely many points. Further, let c, d be positive constants.

Then for any given θ ∈ [0,2π)\F, there are a constant τ, and positive constants R=R(θ) andM =M(θ) such that when |z|=r > R,

(A1) |A(re^iθ)| ≥exp{cδ_A(θ)r^d} if δ_A(θ)>0, (A2) |A(re^iθ)| ≤M r^τ if δ_A(θ)<0,

where τ < 2(ρ(A)−1). Especially, for any θ ∈ [0,2π)\F with δ_A(θ) > 0, there existsl =l(θ)>0 such that sup{R(˜θ),θ˜∈(θ−l, θ+l)}<∞. We call such A(z) as a function of exponential growth and denote the set of all these functions byA. It is also worth noting that there exists τ > −1 for ρ(A) > 1/2, and when ρ(A) > 1, we can even take positive τ. The growth type (A1) must always exist because A(z) is transcendental. Clearly, e^p(z) belongs to A, and A contains many elements from the class of exponential polynomials, which consists of functions of the formm

j=1P_j(z)e^αjz, where P_j(z) are nonzero polynomials and α_j are distinct constants. The study of this class of functions is a classical topic in complex analysis [4, Chapter 3].

If there really exists one θ such that δ_A(θ) < 0 and τ < 0, then in this case, it follows from [3, Corollary] again that there is no Baker wandering domains for suchA(z)∈ A.

Example 1. Letnbe a positive integer andp(z) = α_nzⁿ+· · ·+α₀be a polynomial of degree n. We take A(z) = a_k(z)e^kp(z)+· · ·+a_s(z)e^sp(z), which is a polynomial ine^p(z) of degree k with polynomial coefficients, and 0< s≤k. Set

δ(p, θ) = Re(α_n) cos(nθ)−Im(α_n) sin(nθ).

It is known that e^p(z)= exp{δ(p, θ)|z|ⁿ+O(|z|ⁿ⁻¹)}. Define δ_A(θ) = max{kδ(p, θ), sδ(p, θ)}.

Thus, it is clear that A satisfies A1. Then δ_A(θ) = sδ(p, θ) for θ with δ_A(θ) < 0 and further, for any τ ∈R, we have

|A(re^iθ)| ≤r^degasexp{s

2δ(p, θ)rⁿ} ≤r^τ, as r→ ∞. It follows thatA satisfies A2, consequently, A ∈ A.

Example 2. Suppose that A(z) =b₁(z)e^p1(z)+b₂(z)e^p2(z) wherep_i(z) =α_k,iz^k+

· · ·,(i = 1,2) are two polynomials of degree k > 0, and b_i(z)(i = 1,2) are also polynomials. Now we take δ_A(θ) = max{δ(p₁, θ), δ(p₂, θ)}. When α_k,1/α_k,2 is not negative, we know thatG:={θ, δ(p_i, θ)<0(i= 1,2)} =∅.Thus for θ ∈G, both δ(p₁, θ) and δ(p₂, θ) are negative, so it follows that for any τ ∈R,

|A(re^iθ)| ≤r^degb1exp{1

2δ(p₁, θ)rⁿ}+r^degb2exp{1

2δ(p₂, θ)rⁿ} ≤r^τ, asr tends to ∞.

Example 3. Recall Mittag-Leffler function E_α(z) =

∞ n=0

zⁿ

Γ(α⁻¹n+ 1), 0< α <∞, has the uniform asymptotic behavior [7, Chapter 1, (5.40)],

E_α(z) =

⎧⎨

⎩

αexp (z^α) +O

|z|⁻¹

, |arg(z)| ≤ π 2α, O

|z|⁻¹

, π

2α ≤ |arg(z)| ≤π.

Clearly,A(z) =E_α(z)∈ A with α >1/2, whereδ_A(θ) =−1 for θ ∈(_2α^π ,2π− _2α^π ), and δ_A(θ) = cos(αθ) for [0,2π)\(_2α^π ,2π− _2α^π ).

For solutions of complex differential equations with rational coefficients, Zheng [22] obtained that every transcendental meromorphic function satisfying linear dif- ferential equation with rational coefficients must have no Baker wandering domains and its Julia set has an unbounded component.

As for complex differential equations with transcendental coefficients, Huang and Wang [11] recently considered linear differential equations

f+A(z)f = 0, (1.1)

where entire coefficientA(z) has radial growth behavior similar to functions in our family A, but A(z) decays to zero exponentially in some rays argz = θ. They proved thatE(z) have no Baker wandering domains whereE(z) = f₁f₂, andf₁, f₂ are two linearly independent solutions of (1.1).

In this paper, we will apply the generalized Fuchs’ small arcs lemma [16, Lemma 5] to prove that E(z) to (1.1) with A∈ A and every nonzero solution to

f+A(z)f =H(z), A(z)∈ A, (1.2)

whereH is entire withρ(H)< ρ(A), are of infinite lower order. Based on this fact, we further consider the non-existence of Baker wandering domains for the solutions and the product E, even for their derivatives and primitives.

Note that the lower order can be quite different from the order, since there even exists an entire function of order ρ=∞ and lower order μ= 0, see [7, p.238]. As we know, there are only some attempts to associate the lower order with complex differential equations, see [14], and references therein. We feel a need to state the results on the lower order separately.

Theorem 1. The lower order of any non-zero solution f of (1.2) is infinite.

Theorem 2. The lower order ofE is infinite, whereE(z) =f₁(z)f₂(z), andf₁, f₂ are linearly independent solutions of (1.1) withA∈ A.

Next, with these two theorems in hand, we will prove the non-existence of Baker wandering domains for the solutions f to (1.2) and E to (1.1). Before stating our results, we use f⁽ⁿ⁾(n ∈ Z) to denote the derivatives for n ∈ N and the anti- derivatives for −n∈N, respectively, and f⁽⁰⁾ =f.

Theorem 3. If there exists one θ ∈ F such that δ_A(θ) < 0, then every non-zero solution f of (1.2), along with its f⁽ⁿ⁾(n∈Z), have no Baker wandering domains.

Theorem 4. If there exists one θ ∈F such thatδ_A(θ)<0, then allE⁽ⁿ⁾(z)(n∈Z) have no Baker wandering domains, whereE(z) =f₁(z)f₂(z), andf₁f₂ are linearly independent solutions of (1.1) with A∈ A.

Theorems 1-4 cover a broader range of differential equations than that in [11, Theorem 1.1]. Besides the two examples given in [11], we have another two exam- ples to illustrate some equations that can occur in Theorems 1-4.

Example 4.Letm be any positive integer and f₀ = exp(z^m−1+ ₀^ze^−tmdt). Then μ(f₀) =∞ and f₀ satisfies the equation (1.1) with

−A(z) = (m−1)(m−2)z^m−3+(m−1)²z^2(m−2)+[2(m−1)z^m−2−mz^m−1]e^−zm+e^−2zm. Clearly, ρ(A) =m, and δ_A(θ) = −cos(mθ). Thus, for θ with δ_A(θ)<0,

|A(re^iθ)| ≤2(m−1)²r^2(m−2), for all large r.

Also, 2(m−2)<2(ρ(A)−1). Thus, A∈ A.

Example 5.The function f₀ = exp(ae^z2 +bz) satisfies the equation f−(4a²z²e^2z2 + 2a(2z² + 2bz+ 1)e^z2 +b²)f = 0.

2. lower order of the solutions and E(z) To prove Theorems 1 and 2, we need the following lemmas.

Lemma 1. [11, Lemma 2.7] Let g(z) be an entire function, and ν(r, g) be the central index of g. Then

ρ(g) = lim sup

r→∞

log⁺ν(r, g)

logr , μ(g) = lim inf

r→∞

log⁺ν(r, g) logr .

Before introducing a version of Fuchs’ small-arc lemma, we need recall the upper and the lower logarithmic densities of a given set G ⊂ [0,∞). The logarithmic measure ofGism_l(G) = _G(1/t)dt. The upper and the lower logarithmic densities of Gare defined by

logdens(G) = lim sup

r→∞

m_l(G∩[1, r])

logr , logdens(G) = lim inf

r→∞

m_l(G∩[1, r]) logr . It is easy to see 0≤logdens(G)≤logdens(G)≤1.

The following lemma is a routine consequence of Fuchs’ small-arc lemma [6, Lemma 1], the version here could be found in [16, Lemma 5].

Lemma 2. Let g(z) be a non-constant meromorphic function with finite lower order and let 0 < η < 1. Then there exists a positive constant L and a subset G_η ⊆ [0,∞) of upper logarithmic density at least 1 −η such that if r ∈ G_η is sufficiently large andI_r is a subinterval of [0,2π] of length m then

r

I_r

g(re^iθ) g(re^iθ)

dθ < Lmlog2πe m

T(r, g).

Lemma 3. Let f(z) be an entire function of finite lower order μ, M(r, f) =

|f(re^iθr)| for every r, and η ∈ (0,1). Given s ∈ (0,1), there exist a constant l₀ ∈(0,1/2) and a set F_η with logdens(F_η)>1−η such that

e^−5πM(r, f)^1−s ≤ |f(re^iθ)|, (2.1) for sufficiently larger ∈F_η and all θ such that |θ−θ_r| ≤l₀.

Proof. Following the argument in [18, Lemma 2.4] and Lemma 2, we obtain logM(r, f)−Lmlog2πe

m

T(r, f)≤ |logf(re^iθ)|+ 2π ≤log|f(re^iθ)|+ 5π, (2.2)

for large r ∈ F_η with logdens(F_η) ≥ 1−η and |θ−θ_r| ≤ m. Obviously, we can take m so small that Lmlog_2πe

m

< s. At the same time, (2.1) will follow from

(2.2).

Before the next lemma, we recall one more definition: a meromorphic function α(z) is of growth S(r, f) if T(r, α) = o(T(r, f)) as r → ∞, possibly outside a set of r with finite logarithmic measure.

Lemma 4. ([19, Lemma 2.4]) Let f(z) be a non-constant entire function of lower order μ(f) = μ < ∞. Suppose that α_j(z)(j = 1,2,· · · , m) are meromorphic functions of growth S(r, f). Then there exists a setE ⊂(1,∞)of lower logarithmic density 1, such that

log⁺M(r, α_j)

log⁺M(r, f) →0, (j = 1,2,· · · , m) (2.3) holds simultaneously for r∈E, r→ ∞.

Remark 1. Since the entire functionf in Lemma 4 is not constant, it follows that M(r, f) tends to infinity as r→ ∞. Making use of this fact and (2.3) yields

M(r, α_j)

M(r, f)^1−s →0 (2.4)

forr ∈E, r→ ∞, and any arbitrary constants ∈[0,1).

Lemma 5. Suppose f(z) and g(z) are two non-constant meromorphic functions.

If ρ(f)< μ(g), then T(r, f) = o(T(r, g)).

Proof. From the definition of order and lower order, for 0 < ε < (μ(g)−ρ(f))/3, there existsR >0 such that we have

T(r, f)≤r^ρ(f)+ε ≤r^μ(g)−2ε ≤r^−εT(r, g), for r≥R.

This immediately leads to T(r, f) = o(T(r, g)).

Proof of Theorem 1. Suppose first that f is a nonzero solution of (1.2) with μ(f)<∞. It is easy to see ρ(f)≥ρ(A)> ρ(H).

Claim 1. Moreover, we also have μ(f) ≥ ρ(A). Since ρ(H) < ρ(A) = μ(A), Lemma 5 gives

T(r, H) = o(T(r, A)).

We rewrite (1.2) as

f(z)

f(z) =−A(z) + H(z)

f(z). (2.6)

Then by the first fundamental theorem, it is easy to see T(r, A)≤T(r,f

f ) +T(r,H

f )≤T(r, f) + 2T(r, f) +T(r, H) +O(1).

This implies

T(r, A)≤5T(r, f) +o(T(r, f)) +o(T(r, A)),

outside a set of r with finite linear measure. By [13, Lemma 1.1.1], a lemma from real analysis, we know that for α >1, there exists r₀ >0 such that

T(r, A)≤6T(αr, f), r ≥r₀.

This leads that μ(f)≥μ(A) =ρ(A). Then, Claim 1 is proved.

Applying Lemma 4 to H(z) yields that there is a set F₁ with logdens(F₁) = 1 such that for any given constant s∈[0,1)

|z²H(z)|

M(r, f)^1−s →0, for r ∈F₁ → ∞. (2.7) By Gundersen’s estimation for logarithmic derivatives [8], we have

f(z) f(z)

≤B

T(2r, f)

r log²rlogT(2r, f) 2

≤B(T(2r, f))⁴, (2.8) for all z =re^iθ with|z| ∈F₂∪[0,1], wherem_l(F₂)<∞, and for a positive constant B. The Wiman-Valiron theorem (see [12, p. 187-199] and [13, p. 51]) states that there is a setF₃ with m_l(F₃)<∞ such that

f^(k)(z) f(z) =

ν(r, f) z

k

(1 +o(1)), (2.9)

for z satisfying |f(z)| = M(r, f) and |z| = r ∈ F₃. We may take θ_r such that

|f(re^iθr)|=M(r, f). Recalling Lemma 3, there exists l₀ ∈(0,¹₂) and a set F_η with logdens(F_η) = 1−η such that

e^−5πM(r, f)^1−s ≤ |f(re^iθ)|, (2.10) for all sufficiently large r∈F_η and |θ−θ_r| ≤l₀.

Since the characteristic functions of F₁ and F_η satisfy χ_F1∪Fη(t) =χ_F1(t) +χ_Fη(t)−χ_F1∩Fη(t), we obtain

logdens(F₁) + logdens(F_η)−logdens(F₁ ∩F_η)≤logdens(F₁∪F_η)≤1, which implies logdens(F₁ ∩F_η) ≥ 1−η. Thus, we can take a sequence of points z_n=r_ne^iθn satisfying |f(z_n)|=M(r_n, f) and

r_n ∈F_∗ ^def= (F₁∩F_η)\(F₂∪F₃).

Clearly, logdens(F_∗)≥1−η. Passing to a subsequence of {θ_n}, if needed, we may assume that lim_n→∞θ_n=θ₀.

Further we can assume that the sequence {r_n} is increasing and

logr_n+1 ≤r^d/2_n , (2.11)

where d is also from Definition 2. If not, then there exists a sequence {r_n} with r_n → ∞ such that [r_n,exp(r^d/2n )]∩F_∗ =∅. Therefore,

logdens(R\F_∗)≥ lim

n→∞

exp(r^d/2_n ) r_n

dt t

log exp(rn^d/2) = 1.

This is impossible since

1 = logdens(R)≥logdens(F_∗) + logdens(R\F_∗)>2−η.

Summarizing what we have, we obtain the following claim.

Claim 2. There are l₀ >0 and a sequence z_n=r_ne^iθn such that

|f(z_n)|=M(r_n, f), and lim

n→∞θ_n=θ₀,

and further, (2.7)-(2.11) holds for allr_n and any θ ∈(θ_n−l₀, θ_n+l₀).

In the following, we consider two cases on whether A has the growth type (A1) in someθ∈(θ₀−l₀/2, θ₀+l₀/2)\F or not, where F contains finitely many points.

For simplicity, we denoteδ(θ) =δ_A(θ) in the following.

Case 1.First assume that there is noθ ∈(θ₀−l₀/2, θ₀+l₀/2)\F such that (A1) holds in this direction. We can take ˜θ₁ ∈(θ₀−l₀/2, θ₀) and ˜θ₂ ∈(θ₀, θ₀+l₀/2) such that

max{θ₀−θ˜₁,θ˜₂−θ₀} ≤ π 2(ρ(A) + 1). It follows from (A2) that

|A(re^iθ˜j)| ≤M_jr^τ, (j = 1,2).

Taking suitable branch ofz^−τ makes sure thatA(z)z^−τ is analytic in Ω(r₀; ˜θ₁,θ˜₂) = {z : argz ∈(˜θ₁,θ˜₂), |z| ≥r₀}. It is easy to see that for 0< δ <1, we have

|z^−τA(z)| ≤exp{|z|^κ2−δ}, where κ₂ =π/(˜θ₂−θ˜₁)

in Ω(r₀; ˜θ₁,θ˜₂). Applying Phragm´en-Lindel¨of theorem toz^−τA(z), we conclude that

|z^−τA(z)| ≤M˜ := max{M₁, M₂, r₀^−τM(r₀, A)}, forz ∈Ω(r₀; ˜θ₁,θ˜₂), and so we also have

|A(z)| ≤M r˜ ^τ, for z ∈Ω(r₀; ˜θ₁,θ˜₂) (2.12) where Ω(r₀; ˜θ₁,θ˜₂) denotes the closure of Ω(r₀; ˜θ₁,θ˜₂).

Applying (2.7) in (2.6), we obtain that for large enough n, 1

2

ν(r_n, f) r_n

2

≤ |A(r_ne^iθn)|+ |H(z_n)| M(r_n, f). Then by (2.7) and (2.12), it is easy to see

ν(r_n, f)² ≤2 ˜M r^2+τ_n + 2r_n²|H(z_n)|

M(r_n, f) ≤4 ˜M r_n^2+τ,

which impliesμ(f)≤1 +τ /2< μ(A). This contradicts the fact that μ(f)≥μ(A) in Claim 1.

Case 2.Assume that there is an θ^∗ ∈ (θ₀ −l₀/2, θ₀ +l₀/2)\ F such that (A1) holds in this direction. This also means thatδ(θ^∗)>0. Without loss of generality, we assume thatθ^∗ ∈ (θ₀−l₀/2, θ₀]. We will construct another sequence of points from {z_n} by taking z_n^∗ = r_ne^iθn∗ with θ_n^∗ = θ_n−(θ₀ −θ^∗). Here, we can choose l₀ < min{|θ^∗ −θ|, θ ∈ F}, so θ^∗_n ∈ F for sufficiently large n. It follows from the continuity ofδ(θ) outside F that for large enough n,

1

2δ(θ^∗)≤δ(θ^∗_n)≤ 3

2δ(θ^∗). (2.13)

At the same time, from (2.7) and (2.10), it is obvious that H(z_n^∗)

f(z_n^∗)

≤e^5π |H(z_n^∗)|

M(r_n, f)^1−s →0, (2.14) asn → ∞. Then inserting (A1), (2.8) and (2.14) into (2.6) implies

exp1

2c₁δ(θ^∗)r_n^d

≤ |A(r_ne^iθ∗n)| ≤f(z_n^∗) f(z_n^∗)

+|H(z_n^∗)|

|f(z_n^∗)| ≤2B(T(2r_n, f))⁴. Consequently,

1

2c₁δ(θ^∗₀)r_n^d ≤4 logT(2r_n, f) +O(1), as n → ∞. Thus, for any r ∈[r_n, r_n+1],

logT(2r, f)

log(2r) ≥ logT(2r_n, f) log(2r_n+1) ≥

1

6c₁δ(θ₀^∗)r_n^d+O(1) log 2 +rn^d/2

→ ∞. Thusμ(f) =∞ is proved. Therefore the proof of Theorem 1 is completed.

Proof of Theorem 2. We know from [13, p.77] that

−4A= c

E 2

−E E

2

+ 2E

E , (2.15)

where c is the Wronskian determinant of f₁ and f₁, thus c is a constant. Clearly, E(z) must be transcendental entire, so for any given constant s ∈[0,1)

|c|

M(r, E)^1−s →0, (2.16)

asr→ ∞. Applying Wiman-Valiron theory as in (2.9), we obtain 2E

E −E E

2

=

ν(r, E) z

2

(1 +o(1)), (2.17)

for z satisfying |f(z)| = M(r, f) and |z| = r outside an exceptional set of finite logarithmic measure. On the other hand, again by Gundersen’s estimate [8], we have −E(z)

E(z) 2

+ 2E(z) E(z)

≤E(z) E(z)

²+ 2E(z) E(z)

≤C(T(2r, f))⁴, (2.18) for |z|=routside another exceptional set of finite logarithmic measure, and where C is a constant. Then, similar to the proof of Theorem 1, we can conclude that μ(E) =∞.

3. Baker wandering domain of the solutions

In this section, we will use Nevanlinna theory in angular domains along with our results in Section 2 to prove Theorem 3. For the convenience of the reader, we recall some basic definitions here (e.g. see [7, 23]).

Let g(z) be meromorphic in the closure of Ω(α, β) ^def= {z ∈ C : argz ∈ (α, β)} where β−α∈(0,2π]. Define

A_{α, β}(r, g) = ω π

r 1

1 t^ω − t^ω

r^2ω

log⁺|g(te^iα)|+ log⁺|g(te^iβ)|dt t , B_{α, β}(r, g) = 2ω

πr^ω β

α

log⁺|g(re^iθ)|sinω(θ−α)dθ, C_α,β(r, g) = 2

1<|b_ν|<r

1

|b_ν|^ω − |b_ν|^ω r^2ω

sinω(β_ν −α),

whereω =π/(β−α), andb_ν =|b_ν|e^iβν are the poles of g in the closure of Ω(α, β) appearing according to their multiplicities. The Nevanlinna’s angular characteristic of g is defined by

S_{α, β}(r, g) = A_{α, β}(r, g) +B_{α, β}(r, g) +C_α,β(r, g). (3.1) Lemma 6. ([23, Theorem 2.5.1]) Let f(z) be a meromorphic function in Ω(α− ε, β+ε) for ε >0 and 0< α < β <2π. Then there exists a constant K >0 such that

A_{α, β}

r, f f

+B_{α, β}

r,f f

≤K

log⁺S_{α−ε, β+ε}(r, f) + logr+ 1 , for r >1, except for a set with finite linear measure.

Lemma 7. ([22, Corollary 1]) Let f(z) be a transcendental meromorphic function with at most finitely many poles. If J(f) has only bounded components, then for any complex number a, there exists a constant 0< d < 1 and two sequences {r_n} and {R_n} of positive numbers with r_n → ∞ and R_n/r_n → ∞(n → ∞) such that

M(r, a, f)^d≤L(r, a, f), r ∈G,

where M(r, a, f) = max{|f(z)|:|z−a|=r}, L(r, a, f) = min{|f(z)|:|z−a|=r}, and G=∪^∞n=1{r:r_n< r < R_n} which has an infinite logarithmic measure.

Lemma 8. ([10, Satz 1]) Let p_j(x)(j = 1,2,· · · , n) and f(x) be complex-valued functions on the interval [a, b). Suppose that P_j(x)(j = 1,2,· · · , n) and F(x) are continuous, non-negative functions defined on [a, b) such that |p_j(x)| ≤ P_j(x) and

|f(x)| ≤F(x). Let v(x) satisfy the differential equation v⁽ⁿ⁾−

n j=1

p_j(x)v^(n−j) =f(x), and V(x) satisfy the differential equation

V⁽ⁿ⁾− n

j=1

P_j(x)V^(n−j) =F(x).

Moreover, we assume that |v^(k)(a)| ≤ V^(k)(a) for k = 0,1,· · · , n − 1. Then

|v^(k)(x)| ≤V^(k)(x) holds for x∈[a, b).

Proof of Theorem 3. We now assume that g = f⁽ⁿ⁾(n ∈ Z) has a Baker wandering domain, and complete the proof by reduction to absurdity. From [22, Remark (A)], the Julia set of a transcendental meromorphic function with at most finitely many poles has only bounded components if and only if it has a Baker wandering domain. Since g is transcendental entire, so J(g) only has bounded components. Now, by Lemma 7, there exists 0< d <1 such that

|g(z)| ≥M(r, g)^d, r∈G₀, (3.1) where G₀ is a set of infinite logarithmic measure.

Since there is oneθ ∈[0,2π)\F withδ(θ) =δ_A(θ)<0, then by the continuity of δ(θ) outside the exceptional setF, we can findθ₁ < θ₂ such thatδ(θ_j)<0(j = 1,2) and furthermore,θ₂−θ₁ ≤π/(ρ(A)+1). Thus similarly to the the proof of Theorem 1, we have

|A(z)| ≤M r˜ ^τ, for z ∈Ω(r₀;θ₁, θ₂), (3.2) where ˜M > 1 is a constant.

We first consider f⁽ⁿ⁾(z) when n ≤ 0. It follows from (1.2) that g(z) =f⁽ⁿ⁾(z) satisfies the equation

g^(m)+A(z)g^(m−2) =H(z), (3.3) where m = −n + 2. For any fixed θ ∈ [θ₁, θ₂], we set h(r) = g(re^iθ). Then, h^(k)(r) = e^ikθg^(k)(re^iθ) fork ∈N. Substituting this into (3.3) yields

h^(m)+A(re^iθ)e^i2θh^(m−2) =e^imθH(re^iθ). (3.4) By a simple computation for a positive integer l ≥2, we have

exp(r^l)(k)

=p_k(l−1)(r) exp(r^l), k ∈N, (3.5) wherep_k(l−1)(r) is a polynomial of degreek(l−1) in r with the highest order term asl^kr^k(l−1). Set

M₀ ^def= max{M(r₀, g^(k)), k = 0,1,· · · , m−1}, F(r)^def= M₀

p_m(l−1)(r)−M p˜ _{(m−2)(l−1)}(r)r^[τ]+1

exp(r^l).

By the definition of order, there exists sufficiently larger₀ such that for r≥r₀,

|e^miθH(re^iθ)| ≤exp(r^l)≤F(r),

where we takel = [ρ(H)] + [τ] + 2, and [x] is the integer part of x≥0. Define

V(r) =M₀exp(r^l), (3.6)

and it is easy to check that

|h^(k)(r₀)| ≤M₀ ≤M₀exp((r₀)^l)≤V(r₀), (3.7) for k = 0,1,· · · , m−1, and that V(r) defined in (3.6) satisfies the equation

V^(m)−M r˜ ^[τ]+1V^(m−2) =F(r). (3.8)

Then, applying Lemma 8 to equations (3.4) and (3.8), we get

|g(re^iθ)|=|h(r)| ≤V(r) =M₀exp(r^l), (3.9)

for all z = re^iθ ∈ Ω(r₀;θ₁, θ₂), and if needed, we can take θ₂ a little smaller than before.

Specially for n = 0, combining the estimate (3.9) with (3.1) immediately yields M(r, f)^d≤M₀exp(r^l),

which impliesμ(f)≤l, a contradiction.

By using (3.9), we obtain

A_{θ1, θ2}(r, g) = O(r^l−ω), B_{θ1, θ2}(r, g) =O(r^l−ω)

for ω = π/(θ₂ −θ₁), and for all r ≥ r₀. Since f(z) is entire, then C_θ1,θ2(r, g) = 0.

Summarizing the above argument, so for n≤0, we have

S_{θ1, θ2}(r, f⁽ⁿ⁾) =A_θ1,θ2(r, f⁽ⁿ⁾) +B_θ1,θ2(r, f⁽ⁿ⁾) =O(r^l−ω). (3.10) Now we consider f⁽ⁿ⁾(z) when n >0. For any ε >0, Lemma 6 gives

S_{θ1+ε, θ2−ε}

r,f⁽ⁿ⁾ f

≤

n−1

j=0

S_{θ1+ε, θ2−ε}

r,f^(j+1) f^(j)

≤K ⁿ⁻¹

j=0

log⁺S_{θ1, θ2}(r, f^(j)) + logr+ 1

, (3.11)

with an exceptional set F₁ of finite linear measure. Whenn= 1, (3.10) and (3.11) imply

S_{θ1+ε,θ2−ε}

r,f f

≤K

log⁺S_θ1,θ2(r, f) + logr+ 1

=O(logr), forr ∈F₁, which leads to

S_{θ1+ε,θ2−ε}(r, f)≤S_{θ1+ε,θ2−ε}

r,f f

+S_{θ1+ε,θ2−ε}(r, f) =O(r^l−ω), r∈F₁. Thus, by induction, we can conclude that forr ∈F₁ and n >0,

S_{θ1+ε, θ2−ε}

r,f⁽ⁿ⁾ f

=O(logr), S_{θ1+ε, θ2−ε}(r, f⁽ⁿ⁾) = O(r^l−ω). (3.12) On account of this fact and (3.1), we obtain

S_{α, β}(r, g)≥B_{α, β}(r, g) = 2ω πr^ω

β α

log⁺|g(re^iθ)|sinω(θ−α)dθ

≥ 2

πr^ωdlog⁺M(r, g) β

α

sinω(θ−α)dω(θ−α)

= 4d

πr^ω logM(r, g), r ∈G₀\F₁

whereα=θ₁+ε, β =θ₂−ε, andω =π/(θ₂−θ₁−2ε) forn >0, whileα=θ₁, β =θ₂, and ω = π/(θ₂ − θ₁) for n ≤ 0. Substituting (3.10) and (3.12) into the above inequality yields

logM(r, g)≤ r^ω

4dS_{α, β}(r, g) = r^ω

4dS_{α, β}(r, f⁽ⁿ⁾) = O(r^l), r∈G₀\F₁,

which implies μ(g) < ∞. Therefore, μ(f) < ∞ since μ(g) = μ(f⁽ⁿ⁾) = μ(f) for any n ∈ Z (e.g. see [20, Theorem 4.2]). However, Theorem 1 gives μ(f) = ∞. Therefore, f⁽ⁿ⁾ has no Baker wandering domains for any n∈Z.

4. Baker wandering domain of E(z)

First we consider E⁽ⁿ⁾ for any non-negative integer n. Following from Leibniz formula for higher-order derivatives, we have

E⁽ⁿ⁾= n k=0

C_n^kf₁^(n−k)f₂^(k), where C_n^k = n!

k!(n−k)!. (4.1) Recall that (1.1) is a special case of (1.2) with H(z) ≡ 0. From the proof of Theorem 3., there exists one smaller angular domain Ω(α, β)⊂Ω(θ₁, θ₂) such that

S_α,β(r, f^(k)) =O(r^l−ω), for k ∈Z,

for r outside a set of finite logarithmic measure. Combining this with (4.1) leads to

S_α,β(r, E⁽ⁿ⁾) =O(r^2(l−ω)),

for ω = π/(β −α), and for r outside a set of finite linear measure. Similar to the proof of Theorem 3, ifE⁽ⁿ⁾ has a Baker wandering domain, then μ(E⁽ⁿ⁾)<∞ which contradicts μ(E⁽ⁿ⁾) =μ(E) =∞ by Theorem 2. Therefore, for any integer n≥0,E⁽ⁿ⁾ has no Baker wandering domains.

Now we consider g(z) = E⁽ⁿ⁾ for n <0. We know from [13, p. 77] that E+ 4A(z)E+ 2A(z)E = 0,

which implies thatg satisfies the differential equation

g^(m)+ 4A(z)g^(m−2)+ 2A(z)g^(m−3) = 0, (4.2) where m = −n + 3. Similarly to the proof of Theorem 3, we have (3.2) for z ∈ Ω(r₀;θ₁, θ₂). By the estimate for logarithmic derivatives in [8] again, it follows that we can find another two ˜θ₁,θ˜₁ ∈(θ₁, θ₂)\E₁ such that for sufficiently large r,

|A(re^iθ˜j)|=A(re^iθ˜j) A(re^iθ˜j)

|A(re^iθ˜j)| ≤r^ρ(A)+τ, (4.3) where the set E₁ ⊆[0,2π) has linear measure zero, and j = 1,2. Since ˜θ₁ and ˜θ₂ are chosen to be very close to each other, so by the Phragm´en-Lindel¨of principle, there exist two positive constants r₀, M₁ such that

|A(z)| ≤M₁r^ρ(A)+τ ≤M₁r^N, for z ∈Ω(r₀,θ˜₁,θ˜₂), (4.4) whereN = [ρ(A) +τ] + 1.

For any fixed θ ∈[˜θ₁,θ˜₂], we still take h(r) = g(re^iθ), so

h^(m)+ 4A(re^iθ)e^2iθh^(m−2)+ 2A(re^iθ)e^3iθh^(m−3) = 0. (4.6) Thus, we have from (3.2) and (4.4) that

|A(re^iθ)e^2iθ| ≤M r˜ ^[τ]+1, |A(re^iθ)e^3iθ| ≤M₁r^N. (4.7) Recalling (3.5), set M₀ = max{|g^(k)(0)|, k = 0,1,· · · , m−1.}, and

F(r) =M₀

p_mN(r)−4 ˜M r^[τ]+1p_(m−2)N(r)−2M₁r^Np_(m−3)N(r)

exp(r^N+1), we easily see that

V(r) =M₀exp(r^N+1), satisfies the differential equation

V^(m)−4 ˜M r^[τ]+1V^(m−2)−2M₁r^NV^(m−3) =F(r).

Moreover, there exists a r₀ large enough such that 0 < F(r) and |g^(k)(0)| ≤ V^(k)(0), k = 0,1,· · · , m−1.Applying Lemma 8 again leads to

|g(re^iθ)|=|h(r)| ≤V(r) = M₀exp(r^N+1),

for all z ∈Ω(r₀,θ˜₁,θ˜₂). Hence, log⁺|E⁽ⁿ⁾(z)|=O(r^l) in Ω(˜θ₁,θ˜₂), which implies S_{θ1, θ2}(r, E⁽ⁿ⁾) = O(r^l).

IfE⁽ⁿ⁾has a Baker wandering domain, similar to the proof of Theorem 3, we obtain μ(E⁽ⁿ⁾)<∞, which contradicts Theorem 2. Thus, for any negative integern,E⁽ⁿ⁾ also has no Baker wandering domains.

Therefore, E⁽ⁿ⁾ has no Baker wandering domains for every n∈Z, and Theorem 4 is completely proved.

Acknowledgment: The first author was supported in part by the Academy of Finland (Grant No. 286877). The second author was supported by the National Natural Science Foundation of China (Grant No. 11771090) and Natural Sciences Foundation of Shanghai (Grant No. 17ZR1402900).

References

[1] I. N. Baker, Multiply connected domains of normality in iteration theory,Math.Z., 81(1963), 206-214.

[2] I. N. Baker, An entire function which has wandering domains,J. Aust. Math. Soc., Ser. A, 22(1976), 173-176.

[3] I. N. Baker, Wandering domain in the iteration of entire functions, Proc. Lond. Math.

Soc.,(3)49(1984), 563-576.

[4] Carlos A. Berestein and Roger Gay, Complex Analysis and Special Topics in Harmonics Analysis, New York, Springer-Verlag, 1995.

[5] W. Bergweiler, Iteration of meromorphic functions,Bull. Amer. Math. Soc., 29 (1993), 151- 188.

[6] W. Fuchs, Proof of conjecture of G.P´olya concerning gap series, Illinois J. Math, 7(1963), 661-667.

[7] A. A. Goldberg, I. V. Ostrovskii, Value distribution of meromorphic functions, AMS Trans- lations of Mathematical Monographs series, 2008.