Limit Directions of Julia Sets of Entire Functions and Complex Differential Equations

Qin, Zhigao; Long, Jianren; Xiang, Xuxu

doi:https://doi.org/10.1155/2023/4706738

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Research Article | Open Access

Volume 2023 | Article ID 4706738 | https://doi.org/10.1155/2023/4706738

Limit Directions of Julia Sets of Entire Functions and Complex Differential Equations

Zhigao Qin,¹Jianren Long,¹and Xuxu Xiang¹

Academic Editor: Ljubisa Kocinac

Received23 Mar 2023

Revised14 Sept 2023

Accepted28 Oct 2023

Published29 Nov 2023

Abstract

The limiting directions of Julia sets of infinite order entire functions are studied by combining the theory of complex dynamic system and the theory of complex differential equations, in which the lower bound of the measure of limiting direction of Julia set of entire solutions of complex differential equations is obtained.

1. Introduction and Main Results

Let be a complex plane and be a transcendental meromorphic function, where . is denoted by the -th iteration of . The Fatou set of transcendental meromorphic function is the subset of and satisfies of is a normal family. The Julia set of is the complement of in . We all know that is completely invariant under and open set. is closed and nonempty. Some fundamental knowledge about the complex dynamics of meromorphic functions can be found in[1, 2]. We assume that the reader is familiar with the basic results and standard notation of Nevanlinna’s value distribution theory in . It plays a substantial role in our studying, such as , which can be found in [3, 4]. For a function meromorphic in , the order, lower order, and exponent of convergence of zeros of are given byrespectively, where , is the Nevanlinna characteristic of and denotes the number of in , counting multiplicities. If is an entire function, then the Nevanlinna characteristic can be replaced with , where .

For any , the deficiency of with meromorphic function is defined by

If , then is a deficient value of .

The Lebesgue linear measure of a set is denoted by , and the logarithmic measure of a set is defined by .

In the following, we introduce some related results on the limiting directions of Julia set of meromorphic functions. Baker [5] first observed that the Julia set cannot be contained in any finite set of straight lines for a transcendental entire function . However, this is not true for transcendental meromorphic functions. For example, . Qiao [6] introduced the concept of limiting directions of and demonstrated that the Julia set of a transcendental entire function of finite order possesses an infinite number of limiting directions. The limiting direction of is defined as a ray for which there exists an unbounded sequence such that . For further information, please refer to [6, 7]. Denoting

It is evident that is a closed set. We denote its linear measure by .

In [8], Qiao proved that if is a transcendental entire function of finite lower order, then if ; and if . Later, some observations were made for a transcendental meromorphic function by [9, 10], respectively. They obtained that if and , then

Many results have been obtained involving the limiting directions of the Julia set of transcendental meromorphic functions of finite order, for example [1, 7, 11–13] and therein in references.

However, the limiting direction of the Julia set of entire functions of infinite order remains an open problem. Recently, many scholars have studied the limiting direction of Julia set of entire solutions of infinite order by using the theory of differential equations, such as [11, 12, 14–16].

Furthermore, we define the common Julia limiting directions of derivatives and primitives of an entire function bywhere denotes the -th derivative or -th integral primitive of for or , respectively.

In order to introduce our results, some notations are needed.

Let be two constants such that , is the closed set of and is a ray emanating from the origin.

We recall the concept of an accumulation line of the zero sequence of a transcendental meromorphic function in an angular domain , which can be found in [13, 17]. The radial convergence exponent of the zero sequence of at the ray is defined bywhere denotes the number of zeros of in , counting multiplicities. If , then the ray is considered to be an accumulation line of the zero sequence of . This concept can be used to analyze the growth of solutions of differential equations, as described in [18]. The properties of solutions of the following equation (8) are needed in our results.where , which can be found in [19] [Chapter 7.4], see also Lemma 10 in Section 2 below. The properties of solutions of (8) are used to study the growth of solutions of complex differential equations

See [18, 20] for more details.

In [21], the relationship between and is used to study the growth of solutions of (9), in which infinite order solutions of (9) are characterised by the following condition:as outside a set of finite logarithmic measure. Later, in [22], Long et al. changed the condition toas outside a set of zero upper logarithmic density, and got some results, where .

The asymptotic relationships (10) and (11) can also be used to analyze the limiting direction of the Julia set of solutions of complex differential equations. Wang and Chen [16] studied the common limiting directions of the Julia set of solutions of (9).

Theorem 1 (see [16]). Assume that and are entire functions, where is transcendental and as outside a set of finite logarithmic measure, has a finite deficient value a, i.e., . For every nontrivial solution of (9), we haveMoreover, if , then .

Inspired by Theorem 1 and the idea of asymptotic relations (11) in [22], which the asymptotical relationship (11), we investigate the limiting direction of the Julia set of entire solutions of (9) and derive a lower bound estimate for .

Theorem 2. Let be a nontrivial solution of (8), and the number of accumulation lines of zero sequence of is strictly less than . Let , and let be a transcendental entire function that satisfiesas outside a set of zero upper logarithmic density, where . Then, every nontrivial solution of (9) satisfieswhere .

Next, the condition of of Theorem 1 is replaced with the condition (13) and obtains the following result.

Theorem 3. Let be the entire function having a finite deficient value a, i.e., , and let be given as Theorem 2. Then, every nontrivial solution of (9) satisfieswhere .

The next result is related to Borel exceptional value.

Theorem 4. Let is an entire function of finite order having a finite Borel exceptional value, and let be given as Theorem 2. Then, every nontrivial solution of (9) satisfieswhere and .

Remark 5. It is well-known that Borel exception value is a deficient value and the contrary is not true. So Theorems 3 and 4 both are valuable.

2. Preliminary Lemma

In order to prove our results, we need some preliminary results. First of all, we recall Nevanlinna’s characteristic in an angular domain , which can be found in [23] [Chapter 2]. Let be a meromorphic function on . We definewhere , and are the poles of on , appearing with their respective multiplicities. The Nevanlinna angular characteristic of in is defined as follows:and the order of in is defined by

Lemma 6 (see [23]). Let be a meromorphic function on for and . Then,possibly except a set with finite linear measure.

Lemma 7 (see [10]). Let be holomorphic function, where is a hyperbolic domain. If there exists a point such that , then there exists a constant such that for sufficiently small ,

Remark 8. A domain in is called hyperbolic if contains at least three points. For , the hyperbolic distance from to is defined as , where is the hyperbolic density on . It is well-known that if every component of is simply connected, then , see [10] for more details.

The following lemma provides estimates for the logarithmic derivatives of functions that are analytic in an angular domain. First, let us take a look at the definition of R-set, see [24] for detail. Set . Then, is called an R-set if and as . It is clear that is a set of finite linear measure.

Lemma 9 (see [25]). Let for some and , where . Assuming that is an integer, is analytic in with . Choose . Then, for every outside a set of linear measure zero withThere exist and that only depending on and , not depending on , such thatfor all outside an R-set, where and .

In order to recall the properties of solutions of (8), the following definitions are needed. Let be an entire function with finite positive order . If for any , we havethen blows up exponentially in . If for any , one has that

Then, in , decays to zero exponentially.

The following lemma will be employed to establish Theorem 2.

Lemma 10 (see [19]). Let be a nontrivial solution of equation (8). Set and , where and . Then, has the following properties:(i)In each sector , either blows up or decays exponentially to zero;(ii)If decays exponentially to zero in , then it must blow up exponentially in and , but it is possible for to blow up exponentially in many adjacent sectors;(iii)If decays exponentially to zero in , then has at most finitely many zeros in any closed subsector within ;(iv)If blows up exponentially in and , then, for each , has infinitely many zeros in each sector . Moreover, as ,where is the number of zeros of counting multiplicity in .

Remark 11. It is well-known that the set of accumulation rays of the zero sequence of every nontrivial solution of (8) is a subset of , and the number of accumulation rays is less than or equal to .
Next, we will introduce some information on the correlation between Pólya peak and deficient value. Edrei [26] demonstrated that if is an entire function with , then, for any finite where , there is a series of Pólya peaks of order . As proven in [4], [Theorem 1.1.3] and [23], there is a positive, increasing, and unbounded sequence that is outside of an exceptional set with finite logarithmic measure. The main result of [27] is the spread relation on the Pólya peak, which is stated in the following lemma.

Lemma 12 (see [27]). Let be a transcendental meromorphic function with finite lower order and one deficient value . Let be a positive function such that as . Then, for any fixed sequence of Pólya peaks of order , we havewhere is defined by , and for finite ,.

The following two lemmas are related to Borel exceptional value.

Lemma 13 (see [15]). Let be an entire function of finite order with a finite Borel exceptional value . Then, there exist an entire function with and a polynomial of degree such that

Lemma 14 (see [28]). Let , where , with , and . For any given , we introduce open angleswhere . Then there exists a positive number such that, for ,

3. Proofs of Main Theorems

Proof of Theorem 15. Since , if , then it is clear that . We will prove . Since the number of accumulation lines of zero sequence of is strictly less than , there exists at least one ray is not the accumulation line of the zero sequence of A. Without loss of generally, let such that the ray is not an accumulation line of the zero sequence of , which implies that decays to zero exponentially in either or . In fact, if blows up in both and , then by condition (iv) of Lemma 10, on the one handwhich is impossible. Without loss of generality, assume that decays to zero exponentially in sector , . That is, for any , we have thatand . By simple calculations, for any given , there exists , such that for all and , we haveOn the other hand, for any sufficiently large positive constants , we define and . Then for some , if , we havewhich givesSince is transcendental and satisfies (13) outside of , it yields that, for ,By the Heine theorem, there exists an infinite sequence satisfies , such thatSet . It is easy to see that is a monotone-decreasing measurable set. Moreover, set . Then is independent of , thus, by the Monotone Convergence theorem and (37), we getSuppose that . Then there exists an interval such thatThus, every ray is not a limiting direction of Julia set of for some integer depending on . Then there exists an angular domain such thatfor sufficiently large , is a constant depending on . This means that there exist the corresponding and an unbounded Fatou component of (see [29]), such that . Take an unbounded and connected set , thenis analytic. Because is simply connected, so that is hyperbolic and open, then by Remark 8, for any , we have .
Applying Lemma 7 to mapping mentioned above, there exists a positive constant such thatholds for all .
If , noting the fact thatwhere is a constant, and the integral path can be chosen as the segment of a straight line from 0 to because this integral is independent of the path. It follows from (42) and (43) thatholds for all .
Repeating the above discussion times, we can deduce that for , we haveTherefore, from the definition of Nevanlinna angular characteristic, we know thatIf , in view of [23], [Lemma 2.2.1] and Lemma 6, we conclude thatfor . Then,By (42), we get . By using the similar argument times, it yields thatIt follows from (46) and (49), whenever is positive or not,whereThis implies that . By Lemma 9, there exist two constants and such thatholds for outside a R-set.
Combining (9), (33), and (52), we haveholds for outside an R-set and sufficiently large , where is a positive constant. This is impossible, since can be taken sufficiently large and is a finite positive constant. Hence, we obtainHence, Theorem 2 is completely proved. □

Proof of Theorem 16. Since , if , then it is clear that , Let us prove that , we first assume thatUsing Lemma 12 to , we can take an increasing and unbounded sequence such thatwhere , , with being the R-set. Clearly, for , . Let is defined as in the Proof of Theorem 15. Then, . Similarly as in the Proof of Theorem 15, then there exists an interval such thatClearly, (52) holds. By (52) and , where is defined as in the Proof of Theorem 15, we haveholds for outside an R-set and sufficiently large . This is impossible, since can be taken sufficiently large and is a finite positive constant. Hence, we haveTheorem 3 is completely proved.

Proof of Theorem 17. Since , if , then it is clear that , we will prove for . Let be a Borel exceptional value of . According to Lemma 13, there exists an entire function with and a polynomial of such thatwhere , .
Letwhere and . SetBy (61), we haveFor any and , we haveSince , it follows from (60) and Lemma 14, for any sufficiently small ,as and , where and are positive constants.
It is clear thatSo , where is defined as in the Proof of Theorem 15.
Suppose that . By using a similar reason as in the Proof of Theorem 15, there exists an interval such thatIt is clear that (52) holds. By (9), (52), and (65), we haveholds for outside an R-set and sufficiently large . This is impossible, since can be taken sufficiently large and is a finite positive constant. So, we obtainHence, Theorem 4 is completely proved.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research work was supported by the National Natural Science Foundation of China (Grant nos. 12261023 and 11861023), the Foundation of Science and Technology Project of Guizhou Province of China (Grant No. [2018]5769-05), and Postgraduate Research Funding in Guizhou Province (Grant No. QianJiaoHeYJSKYJJ[2021]087).

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Copyright © 2023 Zhigao Qin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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