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Research article

The lower bound on the measure of sets consisting of Julia limiting directions of solutions to some complex equations associated with Petrenko's deviation

  • Received: 03 May 2023 Revised: 24 May 2023 Accepted: 29 May 2023 Published: 19 June 2023
  • MSC : 30D05, 37F10, 37F50

  • In the value distribution theory of complex analysis, Petrenko's deviation is to describe more precisely the quantitative relationship between T(r,f) and logM(r,f) when the modulus of variable |z|=r is sufficiently large. In this paper we introduce Petrenko's deviations to the coefficients of three types of complex equations, which include difference equations, differential equations and differential-difference equations. Under different assumptions we study the lower bound of limiting directions of Julia sets of solutions of these equations, where Julia set is an important concept in complex dynamical systems. The results of this article show that the lower bound of limiting directions mentioned above is closely related to Petrenko's deviation, and our conclusions improve some known results.

    Citation: Guowei Zhang. The lower bound on the measure of sets consisting of Julia limiting directions of solutions to some complex equations associated with Petrenko's deviation[J]. AIMS Mathematics, 2023, 8(9): 20169-20186. doi: 10.3934/math.20231028

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  • In the value distribution theory of complex analysis, Petrenko's deviation is to describe more precisely the quantitative relationship between T(r,f) and logM(r,f) when the modulus of variable |z|=r is sufficiently large. In this paper we introduce Petrenko's deviations to the coefficients of three types of complex equations, which include difference equations, differential equations and differential-difference equations. Under different assumptions we study the lower bound of limiting directions of Julia sets of solutions of these equations, where Julia set is an important concept in complex dynamical systems. The results of this article show that the lower bound of limiting directions mentioned above is closely related to Petrenko's deviation, and our conclusions improve some known results.



    The content of this paper is related to the value distribution theory and complex dynamic theory in complex analysis, the relevant literatures are available at [9,10,25,29]. On the complex plane, let g be an entire function, its order and lower order denote by σ(g) and μ(g) respectively, see reference [25] for details.

    In Nevanlinna theory, for an entire function g, T(r,g) represents its characteristic function and M(r,g) represents its maximum modulus in the domain |z|<r. For these two quantities, they have a certain magnitude relationship. It's well known that, if 0<r<R<+, then M(r,g) and T(r,g) satisfy the inequalities (see [10])

    R+rRrT(R,g)logM(r,g)T(r,g). (1.1)

    If one takes R as a multiple of r and r goes to infinity, these two quantities are very similar. In order to study the exact relationship between these two quantities, Petrenko [18] introduced the deviation of entire function g at and defined it as follows.

    β(,g)=lim infrlogM(r,g)T(r,g)  and  β+(,g)=lim suprlogM(r,g)T(r,g). (1.2)

    Later they were called Petrenko's deviation. Particularly, β(,g)<β+(,g) is likely to hold, as stated in [18, § 4]. As described in [6], for any increasing function ϕ(r) which is convex for logr, there exists an entire function g that satisfies the following similarity relation

    T(r,g)ϕ(r)logM(r,g). (1.3)

    From the above formula (1.3), we know that there should be many entire functions satisfying β+(,g)=β(,g).

    In the following we will give some specific examples of functions with Petrenko's deviations. For any entire function g with finite lower order μ, let's define a variable B(μ) with respect to μ,

    B(μ):={πμsin(πμ),if  0μ<1/2,πμ,if  μ1/2. (1.4)

    It is shown in [18, Theorem 1] that Petrenko's deviation of g satisfies

    B(μ)β(,g)1. (1.5)

    Both inequalities in (1.5) can hold strictly, see details in [6,18]. For example, the well known Airy integral function Ai(z), which is a solution of g(z)zg(z)=0, has lower order 3/2. It's Petrenko's deviation satisfies

    3π2>limrlogM(r,Ai)T(r,Ai)=3π4>1.

    If the upper deviation β+(,g) and the lower deviation β(,g) are equal and finite, then there exists a constant ν(0,1] such that T(r,f) and logM(r,f) satisfy

    T(r,g)νlogM(r,g) (1.6)

    as r(E), where E(0,+) is an exceptional set. For the function g(z)=ez that we're most familiar with, T(r,ez)=rπ and logM(r,ez)=r satisfies (1.6) with ν=1/π. Generally, the exponential polynomials can also satisfy (1.6) for suitable ν as r(E), where E(0,+) is a set of zero density [17].

    Another example is entire function with Fabry gaps. For an entire function represented by a power series, g(z)=n=0anzλn, if it satisfies the gaps condition λnn as n, we call it having Fabry gaps, see [3]. The maximum modulus of such a function is similar to its minimum modulus, that is,

    logM(r,g)logL(r,g), (1.7)

    as r(E), where E(0,+) is a set of zero density and L(r,g)=min|z|=r|g(z)| is the minimum modulus, see [8]. The orders of this kinds of functions are positive [11, p.651]. Consequently, entire functions with Fabry gaps satisfy (1.6) with ν=1 as r(E).

    In 2021, Heittokangas and Zemirni [12] investigated the zeros of solutions of second order complex differential equation

    f+A(z)f=0, (1.8)

    in which the coefficient A(z) satisfies (1.6). In fact, they found the lower bound of the zero convergence exponent of solution to this equation is closely related to Petrenko's deviation of A(z). In the same article, they also proved the growth of every nontrivial solution to

    f+A(z)f+B(z)f=0, (1.9)

    is infinite under the hypothesise that A(z) is associated with Petrenko's deviation and B(z) is non-transcendental growth in some angular domains.

    For the sake of the following statement, we give the condition that an entire function g(z) has an angular domain in which it is non-transcendental growth. Set

    Ξ(g):={θ[0,2π):lim suprlog+|g(reiθ)|logr<} (1.10)

    and

    ξ(g):=12πmeas(Ξ(g)).

    More specifically, the authors of paper [12] was under the condition that

    β(,B)<11ξ(A) (1.11)

    to prove the infinite growth of solutions of (1.9).

    The following content is some introduction about Julia set and the results of its limiting directions. The Julia set is one of two important subjects of the research of complex dynamics of transcendental meromorphic functions, see more details in [4,28]. Below we briefly recall the known results on the limiting direction of Julia set of meromorphic functions. To prevent confusion, we use fn(nN) to denote the n-th iteration of meromorphic function f. The sequence of functions {fn}(nN) forms a normal family in some domains of the complex plane, and such domains are called the Fatou set, denoted as F(f). Its complement on the complex plane is Julia set, denoted as J(f). The focus of our research is J(f), it's well known that J(f) a nonempty closed set. For transcendental entire functions, the distribution of J(f) in the complex plane is so complicated that it cannot be constrained in a finite number of straight lines, see Baker [1]. However, this is not true for transcendental meromorphic functions, such as the Julia set of tanz being the whole real axis.

    Therefore, the research direction began to focus on the distribution of J(f) on rays, and Qiao introduced this concept in [19]. To explain this concept, let's first give a notation for an angular domain,

    Ω(α,β)={zC:argz(α,β)[0,2π)}.

    So, for any θ[0,2π) and any small ε>0, if the set Ω(θε,θ+ε)J(f) is unbounded, then the ray argz=θ is called a limiting direction of J(f). To see how many such rays there are, the researchers introduced a set called Δ(f), which contains all the limiting direction of J(f). Δ(f) has some properties, such as it is closed and measurable, so one can find a lower bound on the measure of Δ(f). The research on this aspect shows that when f has finite lower order μ(f), the lower bound of measΔ(f) is closely related to μ(f), see [19,30]. So what happens if μ(f) is infinite? Later, Huang et al. [13,14] studied the limiting directions of J(f) of nontrivial solutions to some complex differential equations, these solutions have infinite order or lower order, more results have followed, such as [21,22,23,26]. Below we list a few results that are useful for this article.

    In [13] the authors obtained the lower bound of limiting directions of J(f) of nontrivial solutions to linear complex differential equations

    f(n)+A(z)f=0. (1.12)

    Their result showed that the lower bound is related to the order of A(z). In more depth, the authors of paper [21] investigated the common limiting directions of J(f) of nontrivial solutions of high order linear complex differential equation

    f(n)+An1f(n1)++A0f=0, (1.13)

    where A0 is the dominated coefficient function comparing to other coefficients functions. The common limiting direction of J(f) which just mentioned is defined as follows:

    L(f):=nZΔ(f(n)).

    In the above definition, when n0, f(n) denotes n times of the ordinary derivative of f, when n<0, f(n) means |n| times integral operation.

    Combining the concept of Petrenko's deviation with the results of limiting directions of Julia set of solutions to complex differential equations, the authors of article [27] studied the lower bound of common limiting directions of Julia set of solutions to some complex differential equations with coefficient function related to Petrenko's deviation, the two conclusions are stated below.

    Theorem 1.1. [27] Suppose that f is a nontrivial solution to the complex differential equation (1.9), A(z) is an entire function and has non-transcendental growth angular domain such that ξ(A)>0 and B(z) is a transcendental entire function and has a Petrenko's deviation with β(,B)<11ξ(A), then the common limiting direction of Julia set of f satisfies

    meas(L(f))min{2π,2π(1β(,B)+ξ(A)1)}.

    Theorem 1.2. [27] Suppose that the coefficient A(z) of equation (1.12) is a transcendental entire function and satisfies (1.6) as r outside a set G with logdens(G)<1, then the common limiting direction of Julia set of every nontrivial solution f to (1.12) satisfies

    meas(L(f))2πν.

    Moreover, let f1,f2,,fn be a solution base of equation (1.12) and their product be E:=f1f2fn, then the common limiting direction of Julia set of E satisfies

    meas(L(E))2πν.

    While studying the limiting directions of Julia set of solutions to complex differential equations, the limiting direction of Julia set for the solution of difference equation is also concerned, for examples [5,7,16,24]. In order to state the following Theorem 1.3 which is proved in [5], we give some denotations at first. We set

    ¯P(z,f):=P(f(z+c1),f(z+c2),,f(z+cm)),

    where f(z+ci) are some different shift forms of the complex function f(z) with ci(i=1,,m) being distinct complex numbers, these shifts are closely related to the difference of f. To be specific, ¯P(z,f) is a polynomial in m variables with degree less than d,

    ¯P(z,f):=λ=(k1,,km)Λaλmi=1(f(z+ci))ki

    where aλ are constant coefficients and are nonzeros, every element of Λ is a multi-indices and has the form λ=(k1,,km),kiN, and the degree of ¯P(z,f) is maxλΛ{mi=1ki}<d. In addition, the common limiting directions of Julia sets of distinct shifts of f is defined by

    ¯L(f):=ni=1Δ(f(z+ηi)),

    where n is a finite positive integer, and ηi(i=1,2,n) are distinct complex numbers. Chen et al. [5] obtained the following result about the limiting directions of Julia set of solutions to some complex difference equations.

    Theorem 1.3. [5] Suppose that ¯Pi(z,f) (i=0,1,,n) are distinct polynomials of degree less than d, and the coefficients Ai(z) are entire functions. Assume that A0 is transcendental with finite lower order, and it's dominant compared to the other coefficients, that is, T(r,Aj)=o(T(r,A0)) as r, j=1,2,,n. For any nontrivial entire solution f of

    An(z)¯Pn(z,f)++A1(z)¯P1(z,f)=A0(z), (1.14)

    we have meas(¯L(f))min{2π,π/μ(A0)}. Furthermore, if σ(A0)>max{σ(A1),,σ(An)}, then ¯L(f) contains a closed interval [v1,v2] with v2v1min{2π,π/σ(A0)}.

    Motivated by Theorems 1.2 and 1.3, we study the limiting directions of Julia sets of solutions to complex difference equations with a coefficient related to Petrenko's deviation and obtain the following theorem as our first conclusion of this article.

    Theorem 1.4. Suppose that ¯Pj(z,f)(j=1,2,,n) are distinct polynomials with degree less than d, and A0 is a transcendental entire function with finite lower order μ(A0) and satisfies (1.6) as r(G) with logdens(G)<1. The other coefficients Ai(z)(i=1,2,,n) are distinct entire functions satisfying

    λ=max{σ(A1),,σ(An)}<μ(A0).

    Then the common limiting direction of Julia set of every nontrivial entire solution f to (1.14) satisfies meas(¯L(f))2πν.

    In the paper [16], the authors studied a nonlinear differential-difference equation, it belongs to the general form of Tumura-Clunie equation. This equation is shown as below.

    fn(z)+A(z)~P(z,f)=h(z), (1.15)

    where the coefficient functions A(z),h(z) are entire, and ~P(z,f) denotes a polynomial in the derivatives of the shifts of f,

    ~P(z,f):=sj=1aj(z)li=0(f(i)(z+ci))nij,(i,jN)

    where ci are distinct complex numbers and aj(z) are polynomials. deg~P(z,f)=max1jsli=0nij denotes the degree of ~P(z,f), where nij are positive integers. For more details, see [16, Theorems 1.4 and 1.6]. This inspired us to consider a more general nonlinear complex difference equation than (1.15), combined with the coefficient having Petrenko's deviation. In fact, we obtain the following theorem as our second conclusion of this article.

    Theorem 1.5. Suppose that A0(z) is a transcendental entire function and has Petrenko's deviation with (1.6). A1(z) is an entire function and non-transcendental growth in some angulars such that ξ(A1)>0. Moreover, β(,A0)<11ξ(A1). Then the measure of common limiting direction of Julia set of every nontrivial solution f of equation

    ~P2(z,f)+A1(z)~P1(z,f)=A0(z) (1.16)

    satisfies

    meas(~L(f))min{2π,2π(1β(,A0)+ξ(A1)1)},

    where ~L(f):=nZΔ(f(n)(z+η)) and η is any complex number.

    The third conclusion of this article is about the dynamic properties of solutions to nonlinear complex differential equations. Let's first recall the definition of a differential polynomial for an entire function f. This differential polynomial P(z,f) consists of adding a finite number of differential monomials together, that is

    P(z,f):=lj=1aj(z)fn0j(f)n1j(f(k))nkj, (1.17)

    where aj(z) are coefficient functions and all of them are meromorphic, and n0j,n1j,,nkj are non-negative integer powers. The minimum degree of P(z,f) is denoted by

    γP:=min1jl(ki=0nij).

    Wang et al. [23] observed the limiting directions of Julia set of solutions to the following Eq (1.18), they found when F grows fast in a certain direction, that direction is the limiting direction of Julia set of solution f.

    Theorem 1.6. [23] Let n,s be integers. Suppose that the coefficient F(z) is a transcendental entire function with finite lower order, P(z,f) is a differential polynomial with respect to f, its minimum degree satisfies γPs and its coefficients aj(j=1,2,,l) are entire functions with σ(aj)<μ(F). Then, for every nontrivial transcendental entire solution f of the differential equation

    P(z,f)+F(z)fs=0, (1.18)

    we have meas(L(f))min{2π,π/μ(F)}.

    We improve the above Theorem 1.6 by introducing Petrenko's deviation for one coefficient of the equation and assuming the existence of angular domain of non-transcendental growth for the other coefficient. We state our result as follow.

    Theorem 1.7. Let P1(z,f),P2(z,f) be two differential polynomials with respect to f with min{γP1,γP2}s, where the coefficients a1j,a2j(j=1,2,,l) are entire functions such that max{σ(a1j),σ(a2j)}<μ(A0). Suppose that the coefficient A0(z) is a transcendental entire function with finite lower order and satisfies (1.6), A1(z) is an entire function such that ξ(A1)>0 and β(,A0)<11ξ(A1). s is an integer. Let f be a nontrivial transcendental entire solution of differential equation

    P2(z,f)+A1(z)P1(z,f)+A0(z)fs=0. (1.19)

    Then the measure of common limiting direction of Julia set of f satisfying

    meas(L(f))min{2π,2π(1β(,A0)+ξ(A1)1)}.

    The following lemma is crucial, which describes that when f maps an angular domain to a hyperbolic domain with some special property, f grows by no more than a polynomial in a slightly smaller angular domain. In order to better express this lemma, let us first give some explanations and notations. ˆC denotes the extended complex plane and U is a set contained in ˆC. If ˆCU contains at least three points, then we say U is a hyperbolic domain. For any aCU, we define the quantity CU(a):=inf{λU(z)|za|:zU}, where λU(z) is the hyperbolic density on U. In particular, CU(a)1/2 when every component of U is simply connected, see [30].

    Lemma 2.1. ([30,Lemma 2.2]) Suppose that f(z) is an analytic mapping and f:Ω(r0,θ1,θ2)U, where Ω(r0,θ1,θ2):={reiθ:θ(θ1,θ2),rr0}, and U is a hyperbolic domain. If there exists a point aU{} such that CU(a)>0, then there exists a constant d>0 such that, for sufficiently small ε>0, we have

    |f(z)|=O(|z|d), zΩ(r0,θ1+ε,θ2ε),  |z|.

    Because in the process of proving our theorems, we need to use the Nevanlinna theory in the angular domain, so here we list the notations related to it, which are given in accordance with the literatures [9,29]. Suppose that in the closed angular domain ¯Ω(α,β), g(z) is meromorphic. Set w=π/(βα), and bn=|bn|eiβn are poles of g(z) on ¯Ω(α,β) considering their multiplicities. The three main basic notations are

    Aα,β(r,g)=wπr1(1twtwr2w){log+|g(teiα)|+log+|g(teiβ)|}dtt;Bα,β(r,g)=2wπrwβαlog+|g(reiθ)|sinw(θα)dθ;Cα,β(r,g)=21<|bn|<r(1|bn|w|bn|wr2w)sinw(βnα).

    The sum of these three basic notations is the characteristic function in Ω(α,β), that is,

    Sα,β(r,g)=Aα,β(r,g)+Bα,β(r,g)+Cα,β(r,g).

    Moreover, its growth is defined as

    σα,β(g):=lim suprlogSα,β(r,g)logr. (2.1)

    The following result is a lemma for the estimate of the first two basic notations above of the logarithmic derivative of a meromorphic function f in an angular domain, which is related to the logarithm of the characteristic function of f in a larger angular domain.

    Lemma 2.2. ([29, Theorem 2.5.1]) Let f(z) be a meromorphic function on Ω(αε,β+ε) for ε>0 and 0<α<β2π. Then

    Aα,β(r,ff)+Bα,β(r,ff)K(log+Sαε,β+ε(r,f)+logr+1)

    for r>1 possibly except a set with finite linear measure.

    Further, the estimate of the modulus of the logarithmic derivative of the analytic function in the angular domain is obtained from the following lemma. Let's start with a definition of R set, see [15]. B(zn,rn) are a list of disks, that is, B(zn,rn)={z:|zzn|<rn}. If n=1rn< and zn, then n=1B(zn,rn) is called an R-set. Obviously, the linear measure of set {|z|:zn=1B(zn,rn)} is finite.

    Lemma 2.3. ([13, Lemma 7]) Let g(z) be analytic in Ω(r0,α,β) with σα,β(g)< for some r0>0, and z=reiψ,r>r0+1 and αψβ. Suppose that n(2) is an integer, and that choose α<α1<β1<β. Then, for every εj(0,(βjαj)/2)(j=1,2,,n1) outside a set of linear measure zero with

    αj=α+j1s=1εs,  βj=βj1s=1εs,  j=2,3,,n1.

    There exist K>0 and M>0 only depending on g, ε1,,εn1 and Ω(αn1,βn1), not depending on z, such that

    |g(z)g(z)|KrM(sink(ψα))2

    and

    |g(n)(z)g(z)|KrM(sink(ψα)n1j=1sinkεj(ψαj))2

    for all zΩ(αn1,βn1) outside an R-set, where k=π/(βα) and kεj=π/(βjαj)(j=1,2,,n1).

    By the condition λ=max{σ(A1),,σ(An)}<μ:=μ(A0), we define a constant κ:=λ+μ2, a set

    D:={zC:|A0(z)|>erκ,|z|=r}

    and a set of directions of rays

    H(r):={θ[0,2π):z=reiθD}.

    Noting that A0 is an entire function and recalling the definition of Nevanlinna characteristic function in the complex plane, for some r1>0, if r>r1, we obtain

    2πT(r,A0)=H(r)log+|A0(reiθ)|dθ+[0,2π)H(r)log+|A0(reiθ)|dθmeas(H(r))logM(r,A0)+rκ(2πmeas(H(r))). (3.1)

    Consequently,

    2πmeas(H(r))logM(r,A0)T(r,A0)+rκT(r,A0)(2πmeas(H(r))). (3.2)

    Recall that the definition of lower order, we have

    μ(A0)=lim infrlog+T(r,A0)logr,

    and since A0 has Petrenko's deviation with (1.6) outside G, we deduce that, for rG,

    lim infrmeas(H(r))2πν. (3.3)

    Then there exists a sequence {rn}((r1,+)G) with {rn} as n such that

    lim infnmeas(H(rn))2πν. (3.4)

    We define

    Bn:=j=nH(rj),

    where n=1,2,. Obviously, the set Bn is measurable with meas(Bn)2π, and monotone decreasing as n. Their intersection is

    ˜H:=n=1Bn,

    so ˜H is independent of the value r. In view of (3.4) and the monotone convergence theorem [20, Theorem 1.19], it's easy to get

    meas(˜H)=limnmeas(Bn)=limnmeas(n=jH(rj))2πν. (3.5)

    Let's use proof by contradiction to prove our conclusion and suppose that meas(¯L(f))<2πν. Compared with (3.5), there exists an interval (α,β)[0,2π) such that

    (α,β)˜H,(α,β)¯L(f)=.

    Therefor, for any θ(α,β), the ray {z:argz=θ} is not a limiting direction of Julia set of f(z+ηθ) for some complex number ηθ{ηi:i=1,2,,n} depending on θ. Immediately, by the definition of limiting direction of Julia set, for a constant ξθ depending on θ, we have an angular domain Ω(θξθ,θ+ξθ) such that

    (θξθ,θ+ξθ)(α,β),  Ω(r,θξθ,θ+ξθ)J(f(z+ηθ)= (3.6)

    as r(G) sufficiently large. Then, for a larger rθ, the angular domain Ω(rθ,θξθ,θ+ξθ) is contained in an unbounded Fatou component, say Uθ, of F(f(z+ηθ)) (see [2]). Suppose that ΓUθ is unbounded and connected, then the mapping

    f(z+ηθ):Ω(rθ,θξθ,θ+ξθ)CΓ

    is analytic. Obviously, CΓ is a simply connected hyperbolic domain, then by the explanation before Lemma 2.1, we have CCΓ1/2 for any aΓ{}. From Lemma 2.1, the mapping f(z+ηθ) in Ω(rθ,θξθ,θ+ξθ) satisfies

    |f(z+ηθ)|=O(|z|d1) (3.7)

    for zΩ(rθ,θξθ+ε,θ+ξθε), where ε is a small positive number and d1 is a positive number. For the sake of simplicity, set ¯α:=θξθ+ε and ¯β:=θ+ξθε.

    Choosing suitable rθ(>rθ), for zΩ(rθ,¯α+ε,¯βε) we have z+cjηθΩ(rθ,¯α,¯β) (j=1,2,,m). By (3.7), we obtain

    |f(z+cj)|=|f((z+cjηθ)+ηθ)|=O(|z+cjηθ|d1)=O(|z|M0)

    for zΩ(rθ,¯α+ε,¯βε) as |z|, where M0 is a positive constant. Repeating the above argument some times, we can obtain

    |¯Pi(z,f)|K|z|M,(i=1,2,,n) (3.8)

    for zΩ(rθ,¯α+ε,¯βε) as |z|, where K and M are positive constants. From (1.14), we have

    erκn<|A0(rneiθ)|ni=1|Ai(rneiθ)||¯Pi(rneiθ,f)|KerλnrMn, (3.9)

    where zn=rneiθΩ(rθ,α+ε,βε) as rn(G), K and M are two large enough constants. This is impossible since κ>λ. Hence, we obtain meas(¯L(f))2πν.

    By the assumption and the definition of Petrenko's deviation, for ξ(A1)>0, we have

    1β(,A0)<11ξ(A1).

    We choose two constants ε and d satisfying

    0<ε<1β(,A0)(1ξ(A1))

    and

    1>d>22+ε.

    It's easy to get

    2(1d)d<ε<1β(,A0)(1ξ(A1)). (4.1)

    Set

    Id(r):={θ[0,2π):log|A0(reiθ)|(1d)logM(r,A0)}. (4.2)

    Since A0 is entire function, we have

    2πT(r,A0)=Id(r)log+|A0(reiθ)|dθ+[0,2π)Id(r)log+|A0(reiθ)|dθmeas(Id(r))logM(r,A0)+(2πmeas(Id(r)))(1d)logM(r,A0). (4.3)

    In view the definition of Petrenko's deviation, see (1.2), the formula (4.3) can convert into

    lim suprmeas(Id(r))2π(1dβ(,A0)1dd). (4.4)

    Noting that ε and d satisfy (4.1), we can choose an infinite sequence {rn} such that

    meas(Id(rn))2πdβ(,A0)2π(1d)dπε2πdβ(,A0)2πε2πdβ(,A0)2π(1β(,A0)(1ξ(A1)))>2π(1ξ(A1)). (4.5)

    Then we get a lower bound of the measure of Id(rn). Set Dn:=n=jId(rj) and ~Id:=n=1Dn, so ~Id is independent of r. By the same arguments in section 3, we deduce that

    meas(~Id)=limnmeas(Dn)=limnmeas(n=jId(rj))>2π(1ξ(A1)).

    Thus, we estimate the difference between meas(~Id) and 2π(1ξ(A1)) as follows.

    meas(~Id)2π(1ξ(A1))>2πdβ(,A0)2π(1d)dπε2π(1ξ(A1))=2π[ξ(A1)1d(11β(,A0))ε2]>2π[ξ(A1)2+ε2(11β(,A0))ε2]=2π(1β(,A0)+ξ(A1)1)πε(21β(,A0)). (4.6)

    By the arbitrariness of ε and the condition β(,A0)<11ξ(A1), we have

    meas(~Id)2π(1ξ(A1))2π(1β(,A0)+ξ(A1)1)>0. (4.7)

    Now let's prove the conclusion of the theorem by contradiction, on the contrary we assume

    meas(~L(f))<2π(1β(,A0)+ξ(A1)1). (4.8)

    Noting the definition of Ξ(A1), see (1.10), and (4.7), (4.8), we can find an interval (α,β)(0,2π] satisfying

    (α,β)~IdΞ(A1),  (α,β)~L(f)=. (4.9)

    Therefore, for any θ(α,β), the corresponding ray is not a limiting direction of Julia set of f(nθ)(z+η), where nθ is an integer and depends on θ. By the definition of limiting direction of Julia set, we can take an angular domain Ω(θξθ,θ+ξθ) with (θξθ,θ+ξθ)(α,β) satisfying

    Ω(r,θξθ,θ+ξθ)J(f(nθ)(z+η))= (4.10)

    when r(G) is large enough, ξθ is small enough, and η is a complex constant.

    For a corresponding rθ, we have Ω(rθ,θξθ,θ+ξθ)Uθ, where Uθ is an unbounded Fatou component of F(f(nθ)(z+η)) (see [2]). Choose ΓUθ such that Γ is unbounded and connected set, then CΓ is a simply connected hyperbolic domain. Applying Lemma 2.1 to the analytic mapping

    f(nθ)(z+η):Ω(rθ,θξθ,θ+ξθ)CΓ,

    we deduce that

    |f(nθ)(z+η)|=O(|z|d1) (4.11)

    for rθ and zΩ(rθ,θξθ+ε,θ+ξθε), where ε is a sufficiently small positive constant and d1 is a positive constant. Moreover, we can take a smaller angular domain Ω(rθ,θξθ+2ε,θ+ξθ2ε) where rθ>rθ, such that

    z+ciηΩ(rθ,θξθ+ε,θ+ξθε)

    for all zΩ(rθ,θξθ+2ε,θ+ξθ2ε). For simplicity, denote ¯α:=θξθ+2ε and ¯β:=θ+ξθ2ε. Therefore, it yields that

    |f(nθ)(z+ci)|=O(|z|d1) (4.12)

    for all zΩ(rθ,¯α,¯β) as |z|=r, where ci(i=0,1,,l) are complex numbers. Now, there are two cases based on the sign of nθ.

    If nθ>0, by integral we get

    f(nθ1)(z+ci)=z0f(nθ)(ζ+ci)dζ+c, (4.13)

    this integral is path independent since f(z) is entire. By (4.12) and (4.13), we obtain

    |f(nθ1)(z+ci)|=O(|z|d1+1)

    for zΩ(rθ,¯α,¯β). Repeating the above integral we get

    |f(z+ci)|=O(|z|d1+nθ) (4.14)

    for zΩ(rθ,¯α,¯β).

    If nθ<0. From (4.12), we have

    S¯α,¯β(r,f(nθ)(z+ci))=O(1) (4.15)

    and σ¯α,¯β(r,f(nθ)(z+ci))=0. Noting the fact nθ<0 and applying Lemma 2.3 to f(nθ)(z+ci), we obtain

    |f(z+ci)f(nθ)(z+ci)|KrM (4.16)

    when zΩ(rθ,¯α+ε,¯βε) outside an R set, where K and M are two positive constants and ε is a small positive constant. Then by (4.12) and (4.16), it's easy to obtain

    |f(z+ci)|=O(|z|d1+M) (4.17)

    for all zΩ(rθ,¯α+ε,¯βε) outside an R-set.

    By (4.14) and (4.17), for any nθZ in (4.12), we always can get

    Sα,β(r,f(z+ci))=O(1)

    in the corresponding angular domain. When nθ0, α=¯α,β=¯β; and when nθ<0, α=¯α+ε,β=¯βε. This yields that σα,β(r,f(z+ci))=0, then by Lemma 2.3, for every i=1,2,,l and all zΩ(rθ,α+ε,βε) outside an R-set, we can take two constants Ki>0 and Mi>0 such that

    |f(i)(z+ci)f(z+ci)|KirMi. (4.18)

    From (4.17) and (4.18), it's easy to get that, for all zΩ(α+ε,βε) outside an R-set, the modulus of (f(i)(z+ci))nij satisfies

    |(f(i)(z+ci))nij|=|(f(i)(z+ci)f(z+ci))nijfnij(z+ci)|=O(|z|M), (4.19)

    where M is an appropriate constant. Recall the representations of ~Pi(z,f) we can obtain that

    |~Pi(z,f)|sj=1|aj(z)|li=0|(f(i)(z+ci)f(z+ci))nijfnij(z+ci)|=O(|z|M),(i=1,2) (4.20)

    for all zΩ(α+ε,βε) outside an R-set, and an appropriate constant M. From (1.16), we deduce that, for a suitable sequence {rn} with zn=rneiθΩ(rθ,α+ε,βε),

    [M(rn,A0)]1d<|A0(zn)||~P2(zn,f)|+|A1(z)||~P1(zn,f)|=O(rMn) (4.21)

    as rn. This inequality contradicts the transcendence of A0. So our conclusion is valid.

    Since the entire coefficients A0(z),A1(z) of differential equation (1.19) have the same properties as in Theorem 1.5, then the fact in (4.9) also holds. Then, for some integer nθ which depends on θ, every θ(α,β) is not a limiting direction of Julia set of f(nθ). Therefore, we can choose a constant ξθ which depends on θ, so the interval (θξθ,θ+ξθ)(α,β), and the corresponding angular domain Ω(θξθ,θ+ξθ) satisfies

    Ω(r,θξθ,θ+ξθ)J(f(nθ))= (5.1)

    for sufficiently large r(G). Therefore, for a large enough rθ>r, we have Ω(rθ,θξθ,θ+ξθ)Uθ, where Uθ is an unbounded Fatou component of F(f(nθ)) (see [2]). Taking an unbounded and connected curve ΓUθ and noting that CΓ is simply connected hyperbolic domain, then the mapping

    f(nθ):Ω(rθ,θξθ,θ+ξθ)CΓ

    meets the conditions of Lemma 2.1. Consequently, we have

    |f(nθ)(z)|=O(|z|d1) (5.2)

    for all zΩ(rθ,θξθ+ε,θ+ξθε), where ε is a sufficiently small positive constant and d1 is a positive constant. Denote ¯α:=θξθ+ε and ¯β:=θ+ξθε for simplicity.

    If nθ>0, similar as in section 4, we integrate f(nθ) for nθ times and note the modulus in (4.11), then

    |f(z)|=O(|z|d1+nθ) (5.3)

    for zΩ(rθ,¯α,¯β). Thus, the Nevanlinna angular characteristic of f is

    S¯α,¯β(r,f)=O(1). (5.4)

    If nθ<0, from [29, p.49] and Lemma 2.2, we get

    S¯α+ε,¯βε(r,f(nθ+1))S¯α+ε,¯βε(r,f(nθ+1)f(nθ))+S¯α+ε,¯βε(r,f(nθ))O(log+S¯α,¯β(r,f(nθ))+logr)+S¯α+ε,¯βε(r,f(nθ)) (5.5)

    for ε=ε|nθ|. By (4.11) we get S¯α,¯β(r,f(nθ))=O(1). Combining these together, we deduce

    S¯α+ε,¯βε(r,f(nθ+1))=O(logr).

    Repeating the above argument |nθ| times, we obtain

    S¯α+ε,¯βε(r,f)=O(logr). (5.6)

    From (5.4), (5.6) and combining the two cases above, we deduce that Sα,β(r,f)=O(logr), where

    {α=¯α,β=¯β,fornθ0;α=¯α+ε,β=¯βε,fornθ<0. (5.7)

    By the definition of order in angular domain, see (2.1), we have σα,β(r,f)=0. Applying Lemma 2.3 to f in the angular domain Ω(α,β), we can find two constants K>0 and M>0 such that

    |f(n)(z)f(z)|KrM (5.8)

    for nN and all zΩ(α+ε,βε) outside an R-set. We rewrite (1.19) as

    A0(z)=P2(z,f)fs+A1(z)P1(z,f)fs=l2j=1a2j(z)(ff)n21j(f(k)f)n2kjfn20j+n21j+n2kjs+A1(z)l1j=1a1j(z)(ff)n11j(f(k)f)n1kjfn10j+n11j+n1kjs. (5.9)

    From (1.10), (4.2), (5.3), (5.8) and (5.9), we deduce that, for a suitable sequence zn=rneiθΩ(rθ,α+ε,βε),

    [M(rn,A0)]1d<|A0(zn)|O(rMn)(l1j=1|a1j(zn)|+l2j=1|a2j(zn)|) (5.10)

    as rn. Noting the assumption that max{σ(a1j),σ(a2j)}<μ(A0) and A0 is transcendental, the above inequality is not valid.

    By using the Nevanlinna theory in angular domain, three theorems (Theorems 1.4, 1.5 and 1.7) about the lower bounds on the measure of sets consisting of Julia limiting directions of solutions to three corresponding complex equations were proved. The three equations include the differential or difference of entire functions. The feature of this paper is that the coefficients of these equations associated with Petrenko's deviation. The results of this paper show that the lower bounds mentioned above have close relation with Petrenko's deviation. Meanwhile, the results also extend some conclusions in the related literatures referenced by this paper.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by Natural Science Foundation of Henan (No. 232300420129) and the Key Scientific Research Project of Colleges and Universities in Henan Province (No. 22A110004), China. The author would like to thank the referee for a careful reading of the manuscript and valuable comments.

    The author declares no conflicts of interest.



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