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

Authors: Guowei Zhang; Jian Wang; Lianzhong Yang
Journal: Abstract and Applied Analysis
 (2014)

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## 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.


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## Body

## 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 thatfn, 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 fn 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 denoteΩ(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
(1)Δ(f)={θ∈[0,2π)∣J(f)has the radialdistribution with respect toargz=θ}.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]).
LetA(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
(2)f′′+A(z)f′+B(z)f=0
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.
Letf 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.
Letf 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 Theorem2, we immediately have the following.Corollary 3.
Under the hypothesis of Theorem2, one has mes(Δ(f(k)))≥d, where k is a positive integer.Obviously, Theorem1 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
(3)Ω(α,β,r)={z:z∈Ω(α,β),|z|<r};Ω(r,α,β)={z:z∈Ω(α,β),|z|≥r}
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
(4)Aα,β(r,g)=wπ∫1r‍(1tw-twr2w)hhhhhi×{log⁡+|g(teiα)|+log⁡+|g(teiβ)|}dtt;Bα,β(r,g)=2wπrw∫αβ‍log⁡+|g(reiθ)|sinw(θ-α)dθ;Cα,β(r,g)=2∑1<|bn|<r‍(1|bn|w-|bn|wr2w)sinw(βn-α),
where w=π/(β-α) and bn=|bn|eiβn are poles of g(z) in Ω¯(α,β) appearing according to their multiplicities. The Nevanlinna angular characteristic is defined as
(5)Sα,β(r,g)=Aα,β(r,g)+Bα,β(r,g)+Cα,β(r,g).
In particular, we denote the order of Sα,β(r,g) by
(6)σα,β(g)=limsup⁡r→∞log⁡Sα,β(r,g)log⁡r.IfC¯∖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
(7)CW(a)=inf⁡{λW(z)|z-a|:∀z∈W},
where λW(z) is the hyperbolic density on W. It is well known that if every component of W is simply connected, then CW(a)≥1/2.Lemma 4 (see [14, Lemma 2.2]).
Letf(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 CU(a)>0, then there exists a constant d>0 such that, for sufficiently small ε>0, one has
(8)|f(z)|=O(|z|d),z∈Ω(r0,θ1+ε,θ2-ε),|z|⟶∞.The next lemma shows some estimates for the logarithmic derivative of functions being analytic in an angle. Before this, we recall the definition of anR-set; for reference, see [3]. Set B(zn,rn)={z:|z-zn|<rn}. If ∑n=1∞‍rn<∞ and zn→∞, then ∪n=1∞B(zn,rn) is called an R-set. Clearly, the set {|z|:z∈∪n=1∞B(zn,rn)} is of finite linear measure.Lemma 5 (see [16, Lemma 2.2]).
Letz=reiψ, 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 σα,β(g)<∞. Choose α<α1<β1<β. Then, for every εj∈(0,(βj-αj)/2)(j=1,2,…,n-1) outside a set of linear measure zero with
(9)αj=α+∑s=1j-1‍εs,βj=β-∑s=1j-1‍εs,j=2,3,…,n-1,
there exist K>0 and M>0 only depending on g, ε1,…,εn-1, and Ω(αn-1,βn-1) and not depending on z, such that
(10)|g′(z)g(z)|≤KrM(sink(ψ-α))-2,|g(n)(z)g(z)|≤KrM(sink(ψ-α)∏j=1n-1‍sinkεj(ψ-αj))-2
for all z∈Ω(αn-1,βn-1) outside an R-set D, where k=π/(β-α) and kεj=π/(βj-αj)(j=1,2,…,n-1).Lemma 6 (see [18]).
Letg(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 log⁡dens¯E≥1-(μ(g)/α), where E={r∈[0,∞):m(r)>M(r)cos⁡πα}. Also m(r)=inf⁡|z|=rlog⁡|g(z)| and M(r)=sup⁡|z|=rlog⁡|g(z)|.In the above lemma, the upper logarithmic densities ofE⊂[1,∞) are defined by
(11)log⁡dens¯E=limsup⁡r→∞ml(E∩[0,r])log⁡r,
where ml(E)=∫E‍(dt/t) is logarithmic measure of E.Lemma 7 (see [17]).
LetA(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 Rn with Rn→∞, such that the following two inequalities hold:
(12)mes(Fn)≕mes{δ4T(Rn,A)θ∈[0,2π):log⁡|A(Rneiθ)-a|hhhhhhh≤-δ4T(Rn,A)}≥d>0;(13)|B(Rneiφ)|≥exp⁡{Rnμ(B)-ε},φ∈[0,2π),
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 denoteσ(A), μ(B) by σ, μ, respectively. From the proof of Lemma  2.4 in [17], we know that
(14)d≔δ/4(8H0/2π){5+(2log⁡10eL0)/(log⁡(4/3))},
where H0=2σ/(1-2μ) and L0=2+(2/log⁡2)((2μ+1)/(2μ-1)2).Remark 9 (see [17, Remark 2]).
IfB(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 Theorem1.
By Theorem A, we have already known that every nontrivial solutionf 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 Ii=(αi,βi), i=1,2,…,m, satisfying [αi,βi]⊂S and mes(S∖∪i=1mIi)<ζ/4. For the angular domain Ω(αi,βi), it is easy to see that (αi,βi)∩Δ(f)=∅ and Ω(r,αi,βi)∩J(f)=∅ for sufficiently large r. This implies that, for each i=1,2,…,m, there exist the corresponding ri and unbounded Fatou component Ui of F(f) such that Ω(ri,αi,βi)⊂Ui; see [19]. We take an unbounded and connected section Γi of ∂Ui; then the mapping f:Ω(ri,αi,βi)→C∖Γi is analytic. Since we have chosen Γi such that C∖Γi is simply connected, for any a∈Γi∖{∞}, we have CC∖Γi(a)≥1/2. By applying Lemma 4 to f in every Ω(ri,αi,βi), there exists a positive constant d1 such that, for z∈∪i=1mΩ(ri,αi+ε,βi-ε),
(15)|f(z)|=O(|z|d1),as|z|⟶∞,
where 0<ε<min⁡{ζ/(16m),(βi-αi)/8}, i=1,2,…,m. Thus, recalling the definition of Sα,β(r,f), we immediately have that
(16)Sαi+ε,βi-ε(r,f)=O(1),(i=1,2,…,m).
So σαi+ε,βi-ε(r,f) is finite. Therefore, by Lemma 5, there exist two constants M>0 and K>0 such that
(17)|f(s)(z)f(z)|≤KrM,(s=1,2),
for all z∈∪i=1mΩ(ri,αi+2ε,βi-2ε), outside a R-set H. By Lemma 7, for sufficiently large n, we have
(18)mes(Fn)>d-ζ4.
Therefore, we have
(19)mes(Fn∩S)=mes(Fn∖(Δ(f)∩Fn))≥mes(Fn)-mes(Δ(f))>3ζ4>0.
Then, for each n, we have
(20)mes((⋃i=1mIi)∩Fn)=mes(S∩Fn)-mes((S∖⋃i=1mIi)∩Fn)>3ζ4-ζ4=ζ2.
Thus, there exists an open interval Ii0=(α,β)⊂∪i=1mIi⊂S such that, for infinitely many n,
(21)mes(Fn∩(α,β))>ζ2m>0.
Without loss of generality, we can assume that (21) holds for all n.
Leta be a finite deficient value of A(z) with deficiency δ=δ(a,A). From (2), we have the following inequality:
(22)|B(z)|≤|f′′(z)f(z)|+|f′(z)f(z)|(|A(z)-a|+|a|).
In the following, we consider two cases.
Case  1:
0
<
μ
(
B
)
<
1
/
2. By the definition (12) of Fn and (13), there exists a sequence Rn, which is outside a R-set H, with Rn<Rn+1 and Rn→∞ such that, for every n and φ∈Fn∩(α,β), we have
(23)log⁡|A(Rneiφ)-a|≤-δ4T(Rn,A);(24)|B(Rneiφ)|>exp⁡{Rn(1/2)μ(B)}.
From (17), (22), and (23), for every n≥n0, we get
(25)|B(Rneiφ)|≤KRnM(1+exp⁡{-δ4T(Rn,f)}+|a|).
Thus, by (24), we obtain
(26)exp⁡{Rn(1/2)μ(B)}<KRnM(1+exp⁡{-δ4T(Rn,f)}+|a|).
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 log⁡dens¯E=1 such that, for all z satisfying |z|=r∈E, we have
(27)log⁡|B(z)|>22log⁡M(r,B),
where M(r,B)=max⁡|z|=r|B(z)|. It follows from Remark 9 that there exists a sequence Rn, which is contained in E and outside a R-set H, such that (17), (23), and (27) hold for z=Rneiφ, φ∈Fn∩(α,β). From (17), (22), and (23), we obtain (25). Hence, from (25) and (27), we get
(28)M(Rn,B)2/2≤KRnM(1+exp⁡{-δ4T(Rn,f)}+|a|).
Since B(z) is a transcendental entire function, we have
(29)liminf⁡r→∞log⁡M(r,B)log⁡r=+∞.
Therefore, we can easily obtain a contradiction from (28), when n is sufficiently large. Thus, we complete the proof of Theorem 1.Proof of Theorem2.
We know that every nontrivial solutionf of (2) is an entire function with infinite order. We also obtain the assertion by reduction to contradiction. Assume that
(30)mes(Δ(f)∩Δ(f(k)))<d
and so
(31)ξ≔d-mes(Δ(f)∩Δ(f(k)))>0.
We will show that there must exist an open interval(32)I=(α,β)⊂Δ(f(k))c,0<β-α<d
such that
(33)lim⁡n→∞mes(Δ(f)∩Fn∩I)>0,
where Δ(f(k))c≔[0,2π)∖Δ(f(k)) and Fn is as defined in (12). In order to achieve this goal, we will firstly prove the following:
(34)lim⁡n→∞mes(Fn∖Δ(f))=0.
Otherwise, suppose that there is a subseries {rnk} such that
(35)lim⁡k→∞mes(Fnk∖Δ(f))>0;
then there exist θ0∈Δ(f)c and η>0 satisfying
(36)lim⁡k→∞mes((θ0-η,θ0+η)∩(Fnk∖Δ(f)))>0.
Since argz=θ0 is not a radial distribution of J(f), there exists r0>0 such that
(37)Ω(r0,θ0-η,θ0+η)∩J(f)=∅.
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 CC∖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
(38)|f(z)|=O(|z|d1),z∈Ω(r0,θ0-η+ζ,θ0+η-ζ),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhii|z|⟶∞,
where d1 is a positive constant. Recalling the definition of Sα,β(r,f), we immediately get that
(39)Sθ0-η+ζ,θ0+η-ζ(r,f)=O(1).
Thus, σθ0-η+ζ,θ0+η-ζ(r,f) is finite. Therefore, by Lemma 5, there exist constants M>0 and K>0 such that
(40)|f(s)(z)f(z)|≤KrM,(s=1,2),
for all z∈Ω(r0,θ0-η+2ζ,θ0+η-2ζ), outside a R-set H.
Sinceζ can be chosen sufficiently small, from (36), we have
(41)lim⁡k→∞mes((θ0-η+2ζ,θ0+η-2ζ)∩Fnk)>0.
Thus, we can find an infinite series {rnkeiθnk} such that, for all sufficiently large k, (23), (24), and (27) hold when θnk∈(θ0-η+2ζ,θ0+η-2ζ)∩Fnk. 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 Theorem1, we know that
(42)mesΔ(f)≥d.
From Lemma 7, we have, for all sufficiently large n and any positive ε,
(43)mes(Fn)>d-ε.
Combining (34), (42), and (43), it follows that, for all sufficiently large n,
(44)mes(Δ(f)∩Fn)≥d-ξ4,
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 Ij(j=1,2,…,m) satisfying
(45)Ij⊂Δ(f(k))c,mes(Δ(f(k))c∖⋃i=1mIi)<ξ4.
Since, for sufficiently large n,
(46)mes(Δ(f)∩Fn∩(⋃i=1mIi))+mes(Δ(f)∩Fn∩Δ(f(k)))=mes(Δ(f)∩Fn∩(Δ(f(k))∪(⋃i=1mIi)))≥d-ξ2,
we have
(47)mes(Δ(f)∩Fn∩(⋃i=1mIi))≥d-ξ2-mes(Δ(f)∩Fn∩Δ(f(k)))≥d-ξ2-mes(Δ(f)∩Δ(f(k)))=ξ2.
Thus, there exists an open interval Ii0=(α,β)⊂∪i=1mIi⊂Δ(f(k))c such that, for infinitely many sufficiently large n,
(48)mes(Δ(f)∩Fn∩Ii0)≥ξ2m>0.
Then, we prove that (33) holds.
From (33), we know that there are θ0~ and η~>0 such that
(49)(θ0~-η~,θ0~+η~)⊂I,lim⁡n→∞mes(Δ(f)∩Fn∩(θ0~-η~,θ0~+η~))>0.
Then, there exists r0~ such that Ω(r0~,θ0~-η~,θ0~+η~)∩J(f(k))=∅. By similar argument between (37) and (38), for any ζ~>0, ζ~<η~, we have
(50)|f(k)(z)|=O(|z|d2),z∈Ω(r0~,θ0~-η~+2ζ~,θ0~+η~-2ζ~),hhhhhhhhhhhhhhhhhhhhhhhhhhhhiii|z|⟶∞,
where d2 is a positive constant.
FixrNeiθN∈{rneiθn}, and take a rneiθn∈{rneiθn},n>N. Take a simple Jordan arc γ in Ω(r0~,θ0~-η~,θ0~+η~) which connects rNeiθN to rNeiθn along |z|=rN and connects rNeiθn to rneiθn along argz=θn. For any z∈γ, γz denotes a part of γ, which connects rNeiθN to z. Let L(γ) be the length of γ. Clearly,
(51)L(γ)=O(rn),n⟶∞.
By (50), it follows that
(52)|f(k-1)(z)|≤∫γz‍|f(k)(z)||dz|+ck≤O(|z|d2L(γ))+ck≤O(rnd2+1),n⟶∞.
Similarly, we have
(53)|f(k-2)(z)|≤∫γz‍|f(k-1)(z)||dz|+ck-1≤O(rnd2+2),n⟶∞h⋮|f(z)|≤∫γz‍|f′(z)||dz|+c1≤O(rnd2+k),n⟶∞,
where c1,c2,…,ck are constants, which are independent of n. Therefore,
(54)Sθ0~-η~+ζ~,θ0~+η~-ζ~(r,f)=O(1),
By Lemma 5, we know that (40) also holds for all z∈Ω(r0~,θ0~-η~+2ζ~,θ0~+η~-2ζ~), 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
(55)mes(Δ(f)∩Δ(f(k)))≥d.
The proof is complete.



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