On Radial Distribution of Julia Sets of Solutions to Certain Second Order Complex Linear Differential Equations

Abstract

We mainly investigate the radial distribution of the Julia set of entire solutions to a special second order complex linear differential equation, one of the entire coefficients of which has a finite deficient value.

1. Introduction and Main Results

In this paper, we assume the reader is familiar with standard notations and basic results of Nevanlinna’s value distribution theory; see [1–5]. Some basic knowledge of complex dynamics of meromorphic functions is also needed; see [6, 7]. Let 𝑓 be a meromorphic function in the whole complex plane. We use 𝜎(𝑓) and 𝜇(𝑓) to denote the order and lower order of 𝑓, respectively; see [5] for the definitions.

We define that 𝑓^𝑛, 𝑛∈ℕ, denotes the 𝑛th iterate of 𝑓. The Fatou set 𝐹(𝑓) of transcendental meromorphic function 𝑓 is the subset of the plane ℂ where the iterates 𝑓^𝑛 of 𝑓 form a normal family. The complement of 𝐹(𝑓) in ℂ is called the Julia set 𝐽(𝑓) of 𝑓. It is well known that 𝐹(𝑓) is open and completely invariant under 𝑓; 𝐽(𝑓) is closed and nonempty.

We denote Ω(𝛼,𝛽)={𝑧∈ℂ∣arg𝑧∈(𝛼,𝛽)}, where 0<𝛼<𝛽<2𝜋. Given 𝜃∈[0,2𝜋), if Ω(𝜃−𝜀,𝜃+𝜀)∩𝐽(𝑓) is unbounded, for any 𝜀>0, then the ray arg𝑧=𝜃 is called the radial distribution of 𝐽(𝑓). Define Δ𝑓=[)𝑓.𝜃∈0,2𝜋∣𝐽hastheradialdistributionwithrespecttoarg𝑧=𝜃 (1)

Obviously, Δ(𝑓) is closed and so measurable. We use mesΔ(𝑓) to denote the linear measure of Δ(𝑓). Many important results of radial distribution of transcendental meromorphic functions have been obtained, for example, [8–14]. Qiao [10] proved that mesΔ(𝑓)=2𝜋 if 𝜇(𝑓)<1/2 and mesΔ(𝑓)≥𝜋/𝜇(𝑓) if 𝜇(𝑓)≥1/2, where 𝑓(𝑧) is a transcendental entire function of finite lower order. Recently, Huang and Wang [15, 16] considered the radial distribution of the Julia sets of entire solutions to some special linear complex differential equations and obtained the lower bound of them.

In the present paper, we continue and extend the work of Huang and Wang. In fact, we are devoted to investigating the radial distribution of the Julia set of solutions to second order complex differential equations which is studied by Wu and Zhu [17]. One of coefficients of this equation has relation with deficient value. Actually, they proved the following.

Theorem A (see [17]). Let 𝐴(𝑧) be an entire function with finite order having a finite deficient value and let 𝐵(𝑧) be a transcendental entire function with 𝜇(𝐵)<1/2. Then, every nontrivial solution 𝑓 of equation 𝑓^′′+𝐴(𝑧)𝑓^′+𝐵(𝑧)𝑓=0 (2) is of infinite order.

Our main results are about the lower bound of the radial distribution of the Julia set of solutions to (2).

Theorem 1. Let 𝑓 be a nontrivial solution of (2), where 𝐴(𝑧) is an entire function with finite order having a finite deficient value and 𝐵(𝑧) is a transcendental entire function with 𝜇(𝐵)<1/2; then 𝑚𝑒𝑠(Δ(𝑓))≥𝑑>0, where 𝑑 is defined in Remark 8, Section 2.

Furthermore, we study the radial distribution of Julia set of the derivatives of the nontrivial entire solutions of (2). Indeed, we obtain the following results.

Theorem 2. Let 𝑓 be a nontrivial solution of (2), where 𝐴(𝑧) is an entire function with finite order having a finite deficient value and 𝐵(𝑧) is a transcendental entire function with 𝜇(𝐵)<1/2; then 𝑚𝑒𝑠(Δ(𝑓)∩Δ(𝑓^(𝑘)))≥𝑑, where 𝑘 is a positive integer.

By Theorem 2, we immediately have the following.

Corollary 3. Under the hypothesis of Theorem 2, one has 𝑚𝑒𝑠(Δ(𝑓^(𝑘)))≥𝑑, where 𝑘 is a positive integer.

Obviously, Theorem 1 becomes a corollary of Theorem 2, but we will use the result of Theorem 1 when proving Theorem 2.

2. Preliminary Lemmas

At first, we recall the Nevanlinna characteristic in an angle; see [1]. We set Ω=,|𝑧|;Ω=,|𝑧|𝛼,𝛽,𝑟𝑧∶𝑧∈Ω𝛼,𝛽<𝑟𝑟,𝛼,𝛽𝑧∶𝑧∈Ω𝛼,𝛽≥𝑟 (3) and denote by Ω(𝛼,𝛽) the closure of Ω(𝛼,𝛽). Let 𝑔(𝑧) be meromorphic on the angle Ω(𝛼,𝛽), where 𝛽−𝛼∈(0,2𝜋]. Following [1], we define 𝐴_𝛼,𝛽=𝑤𝑟,𝑔𝜋^𝑟₁1𝑡^𝑤−𝑡^𝑤𝑟^2𝑤×log⁺𝑔𝑡𝑒^𝑖𝛼+log⁺𝑔𝑡𝑒^𝑖𝛽𝑑𝑡𝑡;𝐵_𝛼,𝛽𝑟,𝑔=2𝑤𝜋𝑟^𝑤𝛽_𝛼log⁺𝑔𝑟𝑒^𝑖𝜃()𝐶sin𝑤𝜃−𝛼𝑑𝜃;_𝛼,𝛽𝑟,𝑔=2_{1<|𝑏𝑛|<𝑟}1𝑏_𝑛^𝑤−𝑏_𝑛^𝑤𝑟^2𝑤sin𝑤𝛽_𝑛−𝛼, (4) where 𝑤=𝜋/(𝛽−𝛼) and 𝑏_𝑛=|𝑏_𝑛|𝑒^𝑖𝛽𝑛 are poles of 𝑔(𝑧) in Ω(𝛼,𝛽) appearing according to their multiplicities. The Nevanlinna angular characteristic is defined as 𝑆_𝛼,𝛽𝑟,𝑔=𝐴_𝛼,𝛽𝑟,𝑔+𝐵_𝛼,𝛽𝑟,𝑔+𝐶_𝛼,𝛽𝑟,𝑔. (5) In particular, we denote the order of 𝑆_𝛼,𝛽(𝑟,𝑔) by 𝜎_𝛼,𝛽𝑔=limsup_𝑟→∞log𝑆_𝛼,𝛽𝑟,𝑔.log𝑟 (6)

If ℂ⧵𝑊 contains at least three points, where ℂ is the extended complex plane, then 𝑊 is called a hyperbolic domain. For 𝑎∈ℂ⧵𝑊, define 𝐶_𝑊(𝑎)=inf𝜆_𝑊(𝑧)|𝑧−𝑎|∶∀𝑧∈𝑊, (7) where 𝜆_𝑊(𝑧) is the hyperbolic density on 𝑊. It is well known that if every component of 𝑊 is simply connected, then 𝐶_𝑊(𝑎)≥1/2.

Lemma 4 (see [14, Lemma 2.2]). Let 𝑓(𝑧) be an analytic in Ω(𝑟₀,𝜃₁,𝜃₂); let 𝑈 be a hyperbolic domain and 𝑓∶Ω(𝑟₀,𝜃₁,𝜃₂)→𝑈. If there exists a point 𝑎∈𝜕𝑈⧵{∞} such that 𝐶_𝑈(𝑎)>0, then there exists a constant 𝑑>0 such that, for sufficiently small 𝜀>0, one has 𝑓(𝑧)=𝑂|𝑧|^𝑑,𝑧∈Ω𝑟₀,𝜃₁+𝜀,𝜃₂−𝜀,|𝑧|⟶∞. (8)

The next lemma shows some estimates for the logarithmic derivative of functions being analytic in an angle. Before this, we recall the definition of an 𝑅-set; for reference, see [3]. Set 𝐵(𝑧_𝑛,𝑟_𝑛)={𝑧∶|𝑧−𝑧_𝑛|<𝑟_𝑛}. If ∑^∞_𝑛=1𝑟_𝑛<∞ and 𝑧_𝑛→∞, then ∪^∞_𝑛=1𝐵(𝑧_𝑛,𝑟_𝑛) is called an 𝑅-set. Clearly, the set {|𝑧|∶𝑧∈∪^∞_𝑛=1𝐵(𝑧_𝑛,𝑟_𝑛)} is of finite linear measure.

Lemma 5 (see [16, Lemma 2.2]). Let 𝑧=𝑟𝑒^𝑖𝜓, 𝑟₀+1<𝑟, and 𝛼≤𝜓≤𝛽, where 0<𝛽−𝛼≤2𝜋. Suppose that 𝑛(≥2) is an integer and that 𝑔(𝑧) is analytic in Ω(𝑟₀,𝛼,𝛽) with 𝜎_𝛼,𝛽(𝑔)<∞. Choose 𝛼<𝛼₁<𝛽₁<𝛽. Then, for every 𝜀_𝑗∈(0,(𝛽_𝑗−𝛼_𝑗)/2)(𝑗=1,2,…,𝑛−1) outside a set of linear measure zero with 𝛼_𝑗=𝛼+^𝑗−1_𝑠=1𝜀_𝑠,𝛽_𝑗=𝛽−^𝑗−1_𝑠=1𝜀_𝑠,𝑗=2,3,…,𝑛−1, (9) there exist 𝐾>0 and 𝑀>0 only depending on 𝑔, 𝜀₁,…,𝜀_𝑛−1, and Ω(𝛼_𝑛−1,𝛽_𝑛−1) and not depending on 𝑧, such that 𝑔^′(𝑧)𝑔(𝑧)≤𝐾𝑟^𝑀sin𝑘𝜓−𝛼⁻²,𝑔^(𝑛)(𝑧)𝑔(𝑧)≤𝐾𝑟^𝑀sin𝑘(𝜓−𝛼)^𝑛−1_𝑗=1sin𝑘_𝜀𝑗(𝜓−𝛼_𝑗)⁻² (10) for all 𝑧∈Ω(𝛼_𝑛−1,𝛽_𝑛−1) outside an 𝑅-set 𝐷, where 𝑘=𝜋/(𝛽−𝛼) and 𝑘_𝜀𝑗=𝜋/(𝛽_𝑗−𝛼_𝑗)(𝑗=1,2,…,𝑛−1).

Lemma 6 (see [18]). Let 𝑔(𝑧) be an entire function with 0≤𝜇(𝑔)<1. Then, for every 𝛼∈(𝜇(𝑔),1), there exists a set 𝐸⊂[0,∞) such that log𝑑𝑒𝑛𝑠𝐸≥1−(𝜇(𝑔)/𝛼), where 𝐸={𝑟∈[0,∞)∶𝑚(𝑟)>𝑀(𝑟)cos𝜋𝛼}. Also 𝑚(𝑟)=inf_|𝑧|=𝑟log|𝑔(𝑧)| and 𝑀(𝑟)=sup_|𝑧|=𝑟log|𝑔(𝑧)|.

In the above lemma, the upper logarithmic densities of 𝐸⊂[1,∞) are defined by logdens𝐸=limsup_𝑟→∞𝑚_𝑙(𝐸∩[0,𝑟]),log𝑟 (11) where 𝑚_𝑙(𝐸)=∫_𝐸(𝑑𝑡/𝑡) is logarithmic measure of 𝐸.

Lemma 7 (see [17]). Let 𝐴(𝑧) be a meromorphic function with 𝜎(𝐴)<+∞ and let 𝐵(𝑧) be an entire function with 0<𝜇(𝐵)<1/2. If 𝐴(𝑧) has finite deficient value 𝑎 with deficiency 𝛿=𝛿(𝑎,𝐴)>0, then, for any given constant 𝜀>0, there exists a sequence 𝑅_𝑛 with 𝑅_𝑛→∞, such that the following two inequalities hold: 𝑚𝑒𝑠𝐹_𝑛=∶𝑚𝑒𝑠𝜃∈[0,2𝜋)∶log𝐴𝑅_𝑛𝑒^𝑖𝜃𝛿−𝑎≤−4𝑇𝑅_𝑛,𝐴≥𝑑>0;𝐵𝑅_𝑛𝑒^𝑖𝜑≥exp𝑅_𝑛^{𝜇(𝐵)−𝜀},𝜑∈[0,2𝜋), (12)(13) for all sufficiently large 𝑛, where 𝑑 is a constant depending only on 𝜎(𝐴), 𝜇(𝐵), and 𝛿.

Remark 8. For the sake of simplicity, we denote 𝜎(𝐴), 𝜇(𝐵) by 𝜎, 𝜇, respectively. From the proof of Lemma  2.4 in [17], we know that 𝑑∶=𝛿/48^𝐻0/2𝜋5+2log10𝑒𝐿₀/(),log4/3 (14) where 𝐻₀=2𝜎/(1−2𝜇) and 𝐿₀=2+(2/log2)((2𝜇+1)/(2𝜇−1)²).

Remark 9 (see [17, Remark 2]). If 𝐵(𝑧) is an entire function with 𝜇(𝐵)=0, according to Lemma 6, we only need to give an appropriate modification; then Lemma 7 still holds.

3. Proof of Theorems

Proof of Theorem 1. By Theorem A, we have already known that every nontrivial solution 𝑓 of (2) is an entire function of infinite order. We will obtain the assertion by reduction to contradiction. At first, we suppose that mes(Δ(𝑓))<𝑑, so 𝜁∶=𝑑−mes(Δ(𝑓))>0. Since Δ(𝑓) is closed, obviously 𝑆=[0,2𝜋)⧵Δ(𝑓) is open, so it consists of at most countably many open intervals. We can finitely choose many open intervals 𝐼_𝑖=(𝛼_𝑖,𝛽_𝑖), 𝑖=1,2,…,𝑚, satisfying [𝛼_𝑖,𝛽_𝑖]⊂𝑆 and mes(𝑆⧵∪^𝑚_𝑖=1𝐼_𝑖)<𝜁/4. For the angular domain Ω(𝛼_𝑖,𝛽_𝑖), it is easy to see that (𝛼_𝑖,𝛽_𝑖)∩Δ(𝑓)=0/ and Ω(𝑟,𝛼_𝑖,𝛽_𝑖)∩𝐽(𝑓)=0/ for sufficiently large 𝑟. This implies that, for each 𝑖=1,2,…,𝑚, there exist the corresponding 𝑟_𝑖 and unbounded Fatou component 𝑈_𝑖 of 𝐹(𝑓) such that Ω(𝑟_𝑖,𝛼_𝑖,𝛽_𝑖)⊂𝑈_𝑖; see [19]. We take an unbounded and connected section Γ_𝑖 of 𝜕𝑈_𝑖; then the mapping 𝑓∶Ω(𝑟_𝑖,𝛼_𝑖,𝛽_𝑖)→ℂ⧵Γ_𝑖 is analytic. Since we have chosen Γ_𝑖 such that ℂ⧵Γ_𝑖 is simply connected, for any 𝑎∈Γ_𝑖⧵{∞}, we have 𝐶_ℂ⧵Γ𝑖(𝑎)≥1/2. By applying Lemma 4 to 𝑓 in every Ω(𝑟_𝑖,𝛼_𝑖,𝛽_𝑖), there exists a positive constant 𝑑₁ such that, for 𝑧∈∪^𝑚_𝑖=1Ω(𝑟_𝑖,𝛼_𝑖+𝜀,𝛽_𝑖−𝜀), 𝑓(𝑧)=𝑂|𝑧|^𝑑1,as|𝑧|⟶∞, (15) where 0<𝜀<min{𝜁/(16𝑚),(𝛽_𝑖−𝛼_𝑖)/8}, 𝑖=1,2,…,𝑚. Thus, recalling the definition of 𝑆_𝛼,𝛽(𝑟,𝑓), we immediately have that 𝑆_{𝛼𝑖+𝜀,𝛽𝑖−𝜀}𝑟,𝑓=𝑂(1),(𝑖=1,2,…,𝑚). (16) So 𝜎_{𝛼𝑖+𝜀,𝛽𝑖−𝜀}(𝑟,𝑓) is finite. Therefore, by Lemma 5, there exist two constants 𝑀>0 and 𝐾>0 such that 𝑓^(𝑠)(𝑧)𝑓(𝑧)≤𝐾𝑟^𝑀,(𝑠=1,2), (17) for all 𝑧∈∪^𝑚_𝑖=1Ω(𝑟_𝑖,𝛼_𝑖+2𝜀,𝛽_𝑖−2𝜀), outside a 𝑅-set 𝐻. By Lemma 7, for sufficiently large 𝑛, we have mes𝐹_𝑛𝜁>𝑑−4. (18) Therefore, we have mes𝐹_𝑛∩𝑆=mes𝐹_𝑛⧵Δ𝑓∩𝐹_𝑛≥mes𝐹_𝑛−mesΔ𝑓>3𝜁4>0. (19) Then, for each 𝑛, we have mes^𝑚_𝑖=1𝐼_𝑖∩𝐹_𝑛=mes𝑆∩𝐹_𝑛−mes𝑆⧵^𝑚_𝑖=1𝐼_𝑖∩𝐹_𝑛>3𝜁4−𝜁4=𝜁2. (20) Thus, there exists an open interval 𝐼_𝑖0=(𝛼,𝛽)⊂∪^𝑚_𝑖=1𝐼_𝑖⊂𝑆 such that, for infinitely many 𝑛, mes𝐹_𝑛∩>𝜁𝛼,𝛽2𝑚>0. (21) Without loss of generality, we can assume that (21) holds for all 𝑛.
Let 𝑎 be a finite deficient value of 𝐴(𝑧) with deficiency 𝛿=𝛿(𝑎,𝐴). From (2), we have the following inequality: |𝐵(𝑧≤𝑓)|^′′(𝑧)𝑓(𝑧)+𝑓^′(𝑧)𝑓(𝑧)(|𝐴(𝑧)−𝑎|+|𝑎|). (22)
In the following, we consider two cases.
Case  1: 0<𝜇(𝐵)<1/2. By the definition (12) of 𝐹_𝑛 and (13), there exists a sequence 𝑅_𝑛, which is outside a 𝑅-set 𝐻, with 𝑅_𝑛<𝑅_𝑛+1 and 𝑅_𝑛→∞ such that, for every 𝑛 and 𝜑∈𝐹_𝑛∩(𝛼,𝛽), we have log𝐴𝑅_𝑛𝑒^𝑖𝜑𝛿−𝑎≤−4𝑇𝑅_𝑛,𝐴;𝐵𝑅_𝑛𝑒^𝑖𝜑>exp𝑅_𝑛^{(1/2)𝜇(𝐵)}. (23)(24) From (17), (22), and (23), for every 𝑛≥𝑛₀, we get 𝐵𝑅_𝑛𝑒^𝑖𝜑≤𝐾𝑅^𝑀_𝑛−𝛿1+exp4𝑇𝑅_𝑛,𝑓+|𝑎|. (25) Thus, by (24), we obtain exp𝑅_𝑛^{(1/2)𝜇(𝐵)}<𝐾𝑅^𝑀_𝑛−𝛿1+exp4𝑇𝑅_𝑛,𝑓+|𝑎|. (26) Obviously, when 𝑛 is sufficiently large, this is a contradiction.
Case  2: 𝜇(𝐵)=0. By using Lemma 6, there exists a set 𝐸⊂[0,+∞) with logdens𝐸=1 such that, for all 𝑧 satisfying |𝑧|=𝑟∈𝐸, we have |𝐵(𝑧>√log)|22log𝑀(𝑟,𝐵), (27) where 𝑀(𝑟,𝐵)=max_|𝑧|=𝑟|𝐵(𝑧)|. It follows from Remark 9 that there exists a sequence 𝑅_𝑛, which is contained in 𝐸 and outside a 𝑅-set 𝐻, such that (17), (23), and (27) hold for 𝑧=𝑅_𝑛𝑒^𝑖𝜑, 𝜑∈𝐹_𝑛∩(𝛼,𝛽). From (17), (22), and (23), we obtain (25). Hence, from (25) and (27), we get 𝑀𝑅_𝑛,𝐵^√2/2≤𝐾𝑅^𝑀_𝑛−𝛿1+exp4𝑇𝑅_𝑛,𝑓+|𝑎|. (28) Since 𝐵(𝑧) is a transcendental entire function, we have liminf_𝑟→∞log𝑀(𝑟,𝐵)log𝑟=+∞. (29) Therefore, we can easily obtain a contradiction from (28), when 𝑛 is sufficiently large. Thus, we complete the proof of Theorem 1.

Proof of Theorem 2. We know that every nontrivial solution 𝑓 of (2) is an entire function with infinite order. We also obtain the assertion by reduction to contradiction. Assume that mesΔ𝑓∩Δ𝑓^(𝑘)<𝑑 (30) and so 𝜉∶=𝑑−mesΔ𝑓∩Δ𝑓^(𝑘)>0. (31)
We will show that there must exist an open interval 𝐼=𝛼,𝛽⊂Δ𝑓^(𝑘)𝑐,0<𝛽−𝛼<𝑑 (32) such that lim_𝑛→∞mesΔ𝑓∩𝐹_𝑛∩𝐼>0, (33) where Δ(𝑓^(𝑘))^𝑐∶=[0,2𝜋)⧵Δ(𝑓^(𝑘)) and 𝐹_𝑛 is as defined in (12). In order to achieve this goal, we will firstly prove the following: lim_𝑛→∞mes𝐹_𝑛⧵Δ𝑓=0. (34) Otherwise, suppose that there is a subseries {𝑟_𝑛𝑘} such that lim_𝑘→∞mes𝐹_𝑛𝑘⧵Δ𝑓>0; (35) then there exist 𝜃₀∈Δ(𝑓)^𝑐 and 𝜂>0 satisfying lim_𝑘→∞mes𝜃₀−𝜂,𝜃₀+𝜂∩𝐹_𝑛𝑘⧵Δ𝑓>0. (36) Since arg𝑧=𝜃₀ is not a radial distribution of 𝐽(𝑓), there exists 𝑟₀>0 such that Ω𝑟₀,𝜃₀−𝜂,𝜃₀+𝜂∩𝐽𝑓=0/. (37) This implies that there exists an unbounded component 𝑈 of Fatou set 𝐹(𝑓), such that Ω(𝑟₀,𝜃₀−𝜂,𝜃₀+𝜂)⊂𝑈. Take an unbounded and connected set Γ⊂𝜕𝑈; the mapping 𝑓∶Ω(𝑟₀,𝜃₀−𝜂,𝜃₀+𝜂)→ℂ⧵Γ is analytic. Since ℂ⧵Γ is simply connected, then, for any 𝑎∈Γ⧵{∞}, we have 𝐶_ℂ⧵Γ(𝑎)≥1/2. Now, by applying Lemma 4 to 𝑓 in Ω(𝑟₀,𝜃₀−𝜂,𝜃₀+𝜂), for any 𝜁>0, 𝜁<𝜂, we have 𝑓(𝑧)=𝑂|𝑧|^𝑑1,𝑧∈Ω𝑟₀,𝜃₀−𝜂+𝜁,𝜃₀,|𝑧|+𝜂−𝜁⟶∞, (38) where 𝑑₁ is a positive constant. Recalling the definition of 𝑆_𝛼,𝛽(𝑟,𝑓), we immediately get that 𝑆_{𝜃0−𝜂+𝜁,𝜃0+𝜂−𝜁}𝑟,𝑓=𝑂(1). (39) Thus, 𝜎_{𝜃0−𝜂+𝜁,𝜃0+𝜂−𝜁}(𝑟,𝑓) is finite. Therefore, by Lemma 5, there exist constants 𝑀>0 and 𝐾>0 such that 𝑓^(𝑠)(𝑧)𝑓(𝑧)≤𝐾𝑟^𝑀,(𝑠=1,2), (40) for all 𝑧∈Ω(𝑟₀,𝜃₀−𝜂+2𝜁,𝜃₀+𝜂−2𝜁), outside a 𝑅-set 𝐻.
Since 𝜁 can be chosen sufficiently small, from (36), we have lim_𝑘→∞mes𝜃₀−𝜂+2𝜁,𝜃₀+𝜂−2𝜁∩𝐹_𝑛𝑘>0. (41) Thus, we can find an infinite series {𝑟_𝑛𝑘𝑒^{𝑖𝜃𝑛𝑘}} such that, for all sufficiently large 𝑘, (23), (24), and (27) hold when 𝜃_𝑛𝑘∈(𝜃₀−𝜂+2𝜁,𝜃₀+𝜂−2𝜁)∩𝐹_𝑛𝑘. From (22), (23), (24), (27), and (40) and by the same argument as in Cases 1 and 2 in the proof of Theorem 1, we can obtain contradictions. This implies that (34) is valid.
By Theorem 1, we know that mesΔ𝑓≥𝑑. (42) From Lemma 7, we have, for all sufficiently large 𝑛 and any positive 𝜀, mes𝐹_𝑛>𝑑−𝜀. (43) Combining (34), (42), and (43), it follows that, for all sufficiently large 𝑛, mesΔ𝑓∩𝐹_𝑛𝜉≥𝑑−4, (44) where 𝜉 is defined in (31). Since Δ(𝑓^(𝑘)) is closed, clearly Δ(𝑓^(𝑘))^𝑐 is open, so it consists of at most countably open intervals. We can choose finitely many open intervals 𝐼_𝑗(𝑗=1,2,…,𝑚) satisfying 𝐼_𝑗⊂Δ𝑓^(𝑘)𝑐,mesΔ𝑓^(𝑘)𝑐⧵^𝑚_𝑖=1𝐼_𝑖<𝜉4. (45) Since, for sufficiently large 𝑛, mesΔ𝑓∩𝐹_𝑛∩^𝑚_𝑖=1𝐼_𝑖+mesΔ𝑓∩𝐹_𝑛∩Δ𝑓^(𝑘)=mesΔ𝑓∩𝐹_𝑛∩Δ𝑓^(𝑘)∪^𝑚_𝑖=1𝐼_𝑖𝜉≥𝑑−2, (46) we have mesΔ𝑓∩𝐹_𝑛∩^𝑚_𝑖=1𝐼_𝑖𝜉≥𝑑−2−mesΔ𝑓∩𝐹_𝑛∩Δ𝑓^(𝑘)𝜉≥𝑑−2−mesΔ𝑓∩Δ𝑓^(𝑘)=𝜉2. (47) Thus, there exists an open interval 𝐼_𝑖0=(𝛼,𝛽)⊂∪^𝑚_𝑖=1𝐼_𝑖⊂Δ(𝑓^(𝑘))^𝑐 such that, for infinitely many sufficiently large 𝑛, mesΔ𝑓∩𝐹_𝑛∩𝐼_𝑖0≥𝜉2𝑚>0. (48) Then, we prove that (33) holds.
From (33), we know that there are 𝜃₀ and 𝜂>0 such that 𝜃₀−𝜂,𝜃₀+𝜂⊂𝐼,lim_𝑛→∞mesΔ𝑓∩𝐹_𝑛∩𝜃₀−𝜂,𝜃₀+𝜂>0. (49) Then, there exists 𝑟₀ such that Ω(𝑟₀,𝜃₀−𝜂,𝜃₀+𝜂)∩𝐽(𝑓^(𝑘))=0/. By similar argument between (37) and (38), for any 𝜁̃>0, 𝜁̃<𝜂, we have 𝑓^(𝑘)(𝑧)=𝑂|𝑧|^𝑑2,𝑧∈Ω𝑟₀,𝜃₀−𝜂+2𝜁̃,𝜃₀+𝜁̃,|𝑧|𝜂−2⟶∞, (50) where 𝑑₂ is a positive constant.
Fix 𝑟_𝑁𝑒^𝑖𝜃𝑁∈{𝑟_𝑛𝑒^𝑖𝜃𝑛}, and take a 𝑟_𝑛𝑒^𝑖𝜃𝑛∈{𝑟_𝑛𝑒^𝑖𝜃𝑛},𝑛>𝑁. Take a simple Jordan arc 𝛾 in Ω(𝑟₀,𝜃₀−𝜂,𝜃₀+𝜂) which connects 𝑟_𝑁𝑒^𝑖𝜃𝑁 to 𝑟_𝑁𝑒^𝑖𝜃𝑛 along |𝑧|=𝑟_𝑁 and connects 𝑟_𝑁𝑒^𝑖𝜃𝑛 to 𝑟_𝑛𝑒^𝑖𝜃𝑛 along arg𝑧=𝜃_𝑛. For any 𝑧∈𝛾, 𝛾_𝑧 denotes a part of 𝛾, which connects 𝑟_𝑁𝑒^𝑖𝜃𝑁 to 𝑧. Let 𝐿(𝛾) be the length of 𝛾. Clearly, 𝐿𝛾=𝑂𝑟_𝑛,𝑛⟶∞. (51) By (50), it follows that 𝑓^(𝑘−1)(𝑧)≤_𝛾𝑧𝑓^(𝑘)(𝑧)|𝑑𝑧|+𝑐_𝑘≤𝑂|𝑧|^𝑑2𝐿𝛾+𝑐_𝑘≤𝑂𝑟^𝑑2_𝑛⁺¹,𝑛⟶∞. (52) Similarly, we have 𝑓^(𝑘−2)(𝑧)≤_𝛾𝑧𝑓^(𝑘−1)(𝑧)|𝑑𝑧|+𝑐_𝑘−1≤𝑂𝑟^𝑑2_𝑛⁺²⋮𝑓(𝑧,𝑛⟶∞)≤_𝛾𝑧𝑓^′(𝑧)|𝑑𝑧|+𝑐₁≤𝑂𝑟^𝑑2_𝑛^+𝑘,𝑛⟶∞, (53) where 𝑐₁,𝑐₂,…,𝑐_𝑘 are constants, which are independent of 𝑛. Therefore, 𝑆_{𝜃0−̃𝜃𝜂+𝜁,0+̃𝜁𝜂−}𝑟,𝑓=𝑂(1), (54) By Lemma 5, we know that (40) also holds for all 𝑧∈Ω(𝑟₀,𝜃₀−𝜂+2𝜁̃,𝜃₀+𝜂−2𝜁̃), outside a 𝑅-set 𝐻. By applying similar argument as in Cases 1 and 2 in the proof of Theorem 1, we can deduce contradictions. Therefore, it follows that mesΔ𝑓∩Δ𝑓^(𝑘)≥𝑑. (55) The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors wish to express their thanks to the referee for their valuable suggestions and comments. The present investigation was supported by the National Natural Science Foundation of China under Grant nos. 11301008, 11371225, and 11171013 and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant no. 14B110013.