J. LU KODAI MATH. J. 25 (2002), 72–78
ON JULIA DIRECTIONS OF ENTIRE FUNCTIONS OF SMALL ORDER Jin Lu* Abstract In this paper, the relationship between Julia direction and the growth of entire function of order 0 < l a 1=2 is discussed.
1.
Introduction
A ray wðyÞ ¼ fz ¼ re iy : 0 < r < þyg in the complex plane is called a Julia direction of entire function f ðzÞ, if f ðzÞ takes all finite complex numbers infinitely often in any sector containing wðyÞ, with at most one finite exceptional value. So in any sector containing Julia direction wðyÞ of f ðzÞ, the set of roots of equation f ðzÞ ¼ a is unbounded if a is not such a exceptional value, hence f ðzÞ can not tend to y as jzj ! þy in the sector. But if a ray wðyÞ is not a Julia direction of f ðzÞ, what will happen about f ðzÞ? In [2] H. Yoshida conjectured: If f ðzÞ is an entire function of order less than 1=2 and wðyÞ is ray, then either wðyÞ is a Julia direction of f ðzÞ or f ðzÞ tends y as jzj ! þy in some sector containing wðyÞ. In this paper we discuss this question and prove that H. Yoshida’s conjecture is a‰rmative under some conditions. These conditions are di¤erent from the conditions which were discussed in [2] and we also give a estimate about the growth of this kind of functions in a sector that does not contain Julia directions. Our main result is: Theorem 1. Let f ðzÞ be an entire function of order 0 < l a 1=2, and f ðzÞ be regular growth, i.e., its lower order is equal to its order, then either wðyÞ is a Julia direction of f ðzÞ or in some sector G containing wðyÞ, lim
jzj!þy zAG
log logj f ðzÞj ¼l logjzj
holds. In Section 3 of this paper, an interesting corollary of this theorem will be discussed. * Project supported by Chinese NNSF. 2000 Mathematics Subject Classification: Primary 32D35, 30D20. Received June 4, 2001; revised November 19, 2001.
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on julia directions of entire functions 2.
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Notations and lemmas
Lemma 1 ([1, Theorem 3]). Let f ðzÞ be an entire function of lower order 0 < m a 1=2, then there exists an asymptotic path L tending to y, such that lim inf
jzj!þy zAL
log logj f ðzÞj ¼ m: logjzj
Lemma 2 ([5] or [6, Lemma 6.6]). disc jzj a R. Set
Let f ðzÞ be a meromorphic function on the
N ¼ nðR; f ¼ 0Þ þ nðR; f ¼ 1Þ þ nðR; f ¼ yÞ; where nðR; f ¼ aÞ denotes the number of roots of the equation f ðzÞ ¼ a on jzj a R (Counting multiplicity). If the distance from z ¼ 0 to these N points has a positive lower bound d, then there exists a constant C such that for any 0 < r < R, Tðr; f Þ
0Þ. If there exists a sequence fzn g (jzn j ! y) in Wðs=128; s=128Þ such that j f ðzn Þj < KðnÞ; then there exists a constant C which is independent of n and KðnÞ such that for each n, logj f ðzÞj a Cðlog KðnÞ þ 1Þ holds on the set En ¼ fz ¼ jzn je iy : y A ½s=128; s=128g. Proof. It su‰ces to prove the conclusion holds when n is su‰ciently large. Consider the disc Dn ¼ fz : jz zn j a jzn j sinðs=16Þg. It is easy to see that En H Dn and Dn H Wðs; sÞ. Since jz zn j a jzn j sinðs=32Þ holds for any z A En , from Possion-Jensen formula we have for any z A En , jzn j sinðs=16Þ þ jzn j sinðs=32Þ s ; zn ; f m jzn j sin logj f ðzÞj a jzn j sinðs=16Þ jzn j sinðs=32Þ 16 s ; zn ; f a CT jzn j sin 16 s f a ; zn ; a C T jzn j sin þ1 ; 16 ba
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where Tðr; z0 ; f Þ denotes the characteristic function of f ðzÞ on the disc fz : jz z0 j a rg (In this paper C is always a constant, although it maybe changes in each step). Let N be the number of times that f ðzÞ assumes a and b in Wðs; sÞ. When n is su‰ciently large, the distance from zn to these N points is larger than jzn j sinðs=16Þ. Noticing that the disc fz : jz zn jajzn j sinðs=8Þg lies also in Wðs; sÞ, from Lemma 2 we have s f a jzn j sinðs=8ÞðN þ 1Þ ; zn ; T jzn j sin aC 16 ba jzn j sinðs=8Þ jzn j sinðs=16Þ 2ðjzn j sinðs=8ÞÞ 2 ðN þ 1Þ ðjzn j sinðs=8Þ jzn j sinðs=16ÞÞjzn j sinðs=16Þ þ f ðzn Þ a þ1 þ log ba
log
a Cðlog KðnÞ þ 1Þ: Combining these two estimates the conclusion is obtained.
r
Denote MðWðy1 ; y2 ; r1 ; r2 Þ; f Þ ¼ max z A Wðy1 ; y2 ; r1 ; r2 Þ j f ðzÞj, and Mðr; f Þ ¼ maxjzjar j f ðzÞj. Lemma 4. Let f ðzÞ be an entire function, which assumes distinct complex numbers a and b finite times in the sector Wð2s; 2sÞ ðs > 0Þ. If there exists a sequence fzn g ðjzn j ! yÞ such that j f ðjzn jÞj < KðnÞ: Then for any fixed h ð0 < h < 1Þ, there exists a constant C which is independent of n and KðnÞ such that log MðWðs; s; jzn j1h ; jzn jÞ; f Þ a Cjzn j ðp=sÞh fðlogjzn j þ log KðnÞ þ 1Þg: Proof. It su‰ces to prove the conclusion holds when n is su‰ciently large. Consider the transformation tðzÞ ¼
z p=ð4sÞ jzn j p=ð4sÞ z p=ð4sÞ þ jzn j p=ð4sÞ
:
It is easy to see that under this transformation z ¼ jzn j is mapped to t ¼ 0, and Wð2s; 2sÞ is mapped to the disc jtj < 1. Furthermore Wðs; s; jzn j1h ; jzn jÞ is mapped into jtj < t, where t ¼ 1 ð1=4Þjzn jðp=ð2sÞÞh . Since f ðzÞ assumes a and b only at finite number of points on Wð2s; 2sÞ,
on julia directions of entire functions
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hence when n is large enough, these points are mapped outside of the disc jtj < ð1 þ tÞ=2. Set gðtÞ ¼
f ðzðtÞÞ a ; ba
where zðtÞ is the inverse function of tðzÞ. Then gðtÞ dose not assume 0; 1; y on jtj < ð1 þ tÞ=2. When taking R ¼ ð3 þ tÞ=4, r ¼ ð1 þ tÞ=2 and d ¼ ð1 þ tÞ=2 in Lemma 2 we deduce that 1þt C 1 T ;g a log þ logþ jgð0Þj þ 1 ; 2 1t 1t hence ð1 þ tÞ=2 þ t 1þt T ;g ð1 þ tÞ=2 t 2 C 1 þ a þ log jgð0Þj þ 1 : log 1t ð1 tÞ 2
log Mðt; gÞ a
Since 1 t ¼ ð1=4Þjzn jðp=ð2sÞÞh and gð0Þ ¼ ð f ðjzn jÞ aÞ=ðb aÞ we have log MðWðs; s; jzn j1h ; jzn jÞ; f Þ a log Mðt; gÞ þ C a Cfjzn j ðp=sÞh ðlogjzn j þ log KðnÞ þ 1Þg: 3.
r
The proof of Theorem 1
The proof of Theorem 1. Without loss of generality we assume that y ¼ 0, i.e., we consider the ray wð0Þ ¼ fz : arg z ¼ 0g. If wð0Þ is not a Julia direction of f ðzÞ, then there exists a sector Wð2s; 2sÞ and two distinct complex numbers a; b such that f ðzÞ assumes a and b only finite number of times. Now we prove that on the sector Wðs=128; s=128Þ, log logj f ðzÞj ¼l logjzj jzj!þy lim
zAG
holds. If the conclusion of Theorem 1 were not true, then there exists a positive number r ð0 < r < lÞ and a sequence fzn g ðjzn j ! yÞ in Wðs=128; s=128Þ such that logj f ðzn Þj a jzn j r : Hence from Lemma 3 we have logj f ðjzn jÞj a Cfjzn j r þ 1g:
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Take a positive number h such that ðp=sÞh þ r < ð1 3hÞl, from Lemma 4 we can deduce that log MðWðs; s; jzn j1h ; jzn jÞ; f Þ ð1Þ a Cfjzn j ðp=sÞh ðlogjzn j þ jzn j r þ 1Þg a jzn j lð12hÞ holds for n is su‰ciently large. By Lemma 1 there exists an asymptotic path L tending to y, such that for any e > 0 ðe < l lð1 2hÞ=ð1 hÞÞ, logj f ðzÞj b jzj le ; holds for jzj is su‰ciently large.
zAL
Hence
logj f ðzÞj b jzn jðleÞð1hÞ
ð2Þ
holds for z A L and jzn j1h a jzj a jzn j. Obviously, there exists an positive number N 0 in the interval ½ð1=4Þjzn jð1hÞðleÞ ; ð1=2Þjzn jð1hÞðleÞ such that f 0 ðzÞ has no zero points on the level set L n ¼ fz : logj f ðzÞj ¼ N 0 g, hence L n consists of analytic curves. Consider the set Fn ¼ fz : logj f ðzÞj > N 0 and jzj < jzn jg. Let zn0 be a point at which L intersects the circle fz : z ¼ jzn j1h g, and W n be the connected component of Fn that contains zn0 . By maximum modulus principle, W n V fz : z ¼ jzn jg 0 j. Let yt be the part of fz : jzj ¼ tg lying in W n , yðtÞ and tyðtÞ be the angle measure and linear measure of yt respectively. Notice that jzn jðleÞð1hÞ b jzn j lð12hÞ : Hence from (1) and (2) we have ð3Þ
1 ¼ yðtÞ
1 1 a ð2p 2sÞ ; yðtÞ yðtÞ
jzn j1h a t a jzn j;
Let oðz; yt Þ be the harmonic measure of yt with respect to Fn . 117] we have the estimate ( ð ) ð1=2Þjzn j pffiffiffi dt 0 oðzn ; yt Þ a 9 2 exp p ð4Þ : 2jzn j1h tyðtÞ
From [3, 111–
Hence, by the maximum principle we have ( ð ) ð1=2Þjzn j p ffiffi ffi dt logj f ðzn0 Þj a N 0 þ 9 2 exp p log Mðjzn j; f Þ: 2jzn j1h tyðtÞ Noticing that N 0 A ½ð1=4Þjzn jð1hÞðleÞ ; ð1=2Þjzn jð1hÞðleÞ , from (2) we have ( ð ) ð1=2Þjzn j pffiffiffi dt ð1hÞðleÞ jzn j a 18 2 exp p log Mðjzn j; f Þ: 2jzn j1h tyðtÞ
on julia directions of entire functions
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Hence from (3) we have ð ð1=2Þjzn j ð ð1=2Þjzn j p dt dt ap 2p 2s 2jzn j1h t 2jzn j1h tyðtÞ
pffiffiffi a log log Mðjzn j; f Þ logjzn jð1 hÞðl eÞ þ logð18 2Þ;
i.e., ph logjzn j a log log Mðjzn j; f Þ logjzn jð1 hÞðl eÞ 2p 2s pffiffiffi p log 4: þ logð18 2Þ þ 2p 2s Therefore ph log log Mðjzn j; f Þ a lim sup ð1 hÞðl eÞ 2p 2s logjzn j n!y a l ð1 hÞðl eÞ: Let e ! 0 we have ph=ð2p 2sÞ a lh.
Hence
1 p < a l: 2 2p 2s This contradicts to that the order of f ðzÞ is less than or equal to 1=2. The proof of Theorem 1 is completed. In [2] H. Yoshida proved that, if an entire function f ðzÞ satisfies log Mð2r; f Þ @ log Mðr; f Þ, then the set of ray wðyÞ for which y is a limit point of the set Eð0; f Þ ¼ farg zn : f ðzn Þ ¼ 0g is precisely the set of Julia directions of f ðzÞ. In this situation, the order f ðzÞ is 0 (see [4]). It is well known that this conclusion does not hold for general entire functions. But we have Corollary 1. Let f ðzÞ be an entire function of order 0 < l a 1=2, and f ðzÞ be regular growth. a is a complex number. Then for any limit point y of the set Eða; f Þ ¼ farg zn : f ðzn Þ ¼ ag; the ray wðyÞ is a Julia direction of f ðzÞ. Furthermore, if a and b are distinct complex numbers, then the set of ray wðyÞ for which y is the limit point of the set Eða; b; f Þ ¼ farg zn : f ðzn Þ ¼ a or f ðzn Þ ¼ bg is precisely the set of Julia directions of f ðzÞ. Proof. Let y be a limit point of the set Eða; f Þ. If wðyÞ is not a Julia direction of f ðzÞ, then from Theorem 1 there exists a sector G containing wðyÞ
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such that j f ðzÞj tends to þy as jzj ! þy in G. Hence f ðzÞ can not assume a infinite number of times in the sector G. This contradicts to that y is a limit point of the set Eða; f Þ. Hence wðyÞ is a Julia direction of f ðzÞ. Furthermore, if y is a limit point of the set Eða; b; f Þ, by the same discussion as above we can see that wðyÞ is a Julia direction of f ðzÞ. If y is not a limit point of the set Eða; b; f Þ, then there exists a sector G containing wðyÞ such that f ðzÞ assumes a and b at most finite number of times in the sector G. Hence wðyÞ is not a Julia direction of f ðzÞ. r References [ 1 ] K.-H. Chang, Asymptotic values of entire and meromorphic functions, Sci. Sinica, 20 (1977), 720–739. [ 2 ] H. Yoshida, Julia directions of entire functions of smooth growth, Nagoya Math. J., 87 (1982), 41–57. [ 3 ] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. [ 4 ] W. K. Hayman, On Iversen’s theorem for meromorphic functions with few poles, Acta Math., 141 (1978), 115–145. [ 5 ] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin-Heidelberg, 1993. [ 6 ] L. Yang, Value Distribution Theory and New Research on it, Science Press, Beijing, 1982 (in Chinese). Department of Mathematics Fudan University Shanghai 200433 China e-mail:
[email protected]
