JULIA LIMITING DIRECTIONS OF ENTIRE SOLUTIONS OF COMPLEX DIFFERENTIAL EQUATIONS∗

(王珺)

School of Mathematical Sciences,Fudan University,Shanghai 200433,China E-mail:majwang@fudan.edu.cn

Xiao YAO (姚潇)†

School of Mathematical Sciences and LPMC,Nankai University,Tianjin 300071,China E-mail:yaoxiao@nankai.edu.cn

Chengchun ZHANG (张城纯)

School of Mathematical Sciences,Fudan University,Shanghai 200433,China E-mail:18210180014@fudan.edu.cn

Abstract For entire or meromorphic function f,a value θ∈[0,2π)is called a Julia limiting direction if there is an unbounded sequence{zn}in the Julia set satisfyingOur main result is on the entire solution f of P(z,f)+F(z)fs=0,where P(z,f)isadifferential polynomial of f with entire coefficients of growth smaller than that of the entire transcendental F,with the integer s being no more than the minimum degree of all differential monomials in P(z,f).We observe that Julia limiting directions of f partly come from the directions in which F grows quickly.

Key words Julia set;meromorphic function;Julia limiting direction;complex differential equations

1 Introduction and Main Results

There are a lot of works centered around the dynamics of classes of transcendental functions,such as the Speier class and the Eremenko-Lyubich class.This paper is another contribution in this direction,and we focus on transcendental entire solutions of complex differential equations.For these transcendental solutions,we investigate the geometric property of their Julia sets near∞,which is one of the questions in transcendental iteration theory distinct from the iteration of rational functions.

Baker[2]observed that,when

f

is a transcendental entire function,J(

f

)cannot be contained in any finite set of straight lines.However,this is not true for transcendental meromorphic functions,such as J(tan

z

)=R.From the viewpoint of angular distribution,Qiao[11]introduced the limiting direction of the Julia set.A value

θ

∈[0

,

2

π

)is said to be a limiting direction of the Julia set of

f

if there is an unbounded sequence{

z

}⊆J(

f

)such that

For brevity,in this paper we call a limiting direction of the Julia set of

f

a Julia limiting direction of

f

.We denote by

L

(

f

)the set of all Julia limiting directions of

f

;it is a non-empty closed set in[0

,

2

π

)and will reveal the large-scale geometry property of J(

f

).Since any rational function,as well as any polynomial,can be treated as a map between two Riemann spheres,it makes no sense for us to consider the Julia limiting direction for rational functions.Furthermore,we identify[0

,

2

π

)with the circle S:={

z

∈C:|

z

|=1}and intervals in[0

,

2

π

)with arcs on the circle for convenience.Value distribution theory plays an important role in studying transcendental meromorphic functions,and its standard notations as well as its well-known theorems can be found in[7,8].For example,we denote by

T

(

r,f

)and

N

(

r,f

)the characteristic function and the integrated counting function of poles,respectively,with respect to

f

.The order

ρ

(

f

)and the lower order

µ

(

f

)are de fined by

For transcendental entire functions,Qiao[11]noticed a relation between the Lebesgue measure of

L

(

f

)and the growth order of

f

.

Theorem 1.1

([11])Let

f

be a transcendental entire function of lower order

µ<

∞.Then there exists a closed interval

I

⊆

L

(

f

)such that

where meas(

I

)is the Lebesgue measure of the set

I

.The condition that

µ<

∞in Theorem 1.1 is necessary,since Baker[2]proved that there exists an entire function

f

of in finite lower order with a property implying that

L

(

f

)is a single point set.Furthermore,Qiao[11]showed that the estimate in Theorem 1.1 is sharp,which is veri fied by modifying functions in the Mittag-Leffler class.Recalling J(tan

z

)=R,Theorem 1.1 fails for general meromorphic functions,but under some certain conditions,Theorem 1.1 can be generalized;see[12,21]for the details.For entire functions of in finite lower order,what is the sufficient condition for the existence of the lower bound of meas(

L

(

f

))?There are already some considerations regarding solutions of complex differential equations,for example,the linear equations

(see[9]),and the nonlinear equations,such as Riccati equations

(see[20]).We only state the result on linear equations(1.2)here.

Theorem 1.2

([9])Suppose that all coefficients of(1.2)are entire functions of finite lower order,that

a

is transcendental and that

T

(

r,a

)=

o

(

T

(

r,a

))(

i

=1

,

2

,...,n

−1).Then every nonzero solution

f

of(1.2)is of in finite lower order,and

Furthermore,under the hypothesis of Theorem 1.2,there even exists(see[17])

where

f

(

n

∈Z)denote the derivatives for

n

∈N and the anti-derivatives for−

n

∈N,and

f

=

f

.In addition,a corresponding investigation has been done for equations(1.2)with exponential coefficients[16].However,it is not clear what is behind the inequalities(1.1)and(1.4).Recently,we found out that for the entire function

f

,the direction in which

f

grows more quickly than any polynomial is a Julia limiting direction of

f

;see Lemma 2.2 in the next section.We introduce the following concept of transcendental direction to describe these directions in which

f

grows quickly:a value

θ

∈[0

,

2

π

)is said to be a transcendental direction of

f

if there exists an unbounded sequence of{

z

}such that

We use

TD

(

f

)to denote the union of all transcendental directions;clearly

TD

(

f

)is a non-empty compact set in[0

,

2

π

).We will see that Julia limiting directions of solutions to(1.2)partly come from the transcendental directions of the dominanting coefficient

a

.Furthermore,in this paper,we investigate more general differential equations,which even contain some non-linear differential equations.Before stating our results,we first introduce the terminology of differential polynomials of

f

.The differential polynomial

P

(

z,f

)is a finite sum of differential monomials generated by

f

,that is,

where the coefficients

a

(

z

)are meromorphic,and the powers

n

,n

,...,n

are non-negative integers.We use

γ

to denote the minimum degree of

M

as

Theorem 1.3

Suppose that

s,n

are integers,

F

(

z

)is a transcendental entire function of finite lower order,and that

P

(

z,f

)is a differential polynomial in

f

with

γ

≥

s

,where all coefficients

a

(

j

=1

,

2

,...,l

)are polynomials if

µ

(

F

)=0,or all

a

(

j

=1

,

2

,...,l

)are entire functions and

ρ

(

a

)

<µ

(

F

).Then,for every nonzero transcendental entire solution

f

of the differential equation

we have

TD

(

f

)∩

TD

(

F

)⊆

L

(

f

)and

Clearly,when

s

=1

,F

=

a

(

z

)and

P

(

z,f

)=

f

+

a

(

z

)

f

+

...

+

a

(

z

)

f

,we immediately obtain the following corollary from Theorem 1.3:

Corollary 1.4

Suppose that all coefficients of(1.2)are entire functions of finite lower order,that

a

is transcendental and that all

a

(

i

=1

,

2

,...,n

−1)are polynomials if

µ

(

a

)=0,or that all

a

(

i

=1

,

2

,...,n

−1)satisfy

ρ

(

a

)

<µ

(

a

).Then,for every nonzero solution

f

of(1.2),we have

TD

(

f

)∩

TD

(

a

)⊆

L

(

f

)

,k

∈Z and

As for the case that

s

=0,there is another corollary from Theorem 1.3 which can treat not only the non-homogeneous linear equation corresponding to(1.2)but also the non-linear differential equations

P

(

z,f

)=

F

(

z

).

Corollary 1.5

Suppose that

F

and

P

(

z,f

)are de fined as in Theorem 1.3.Then,for every nonzero entire solution

f

of the equation

P

(

z,f

)=

F

(

z

),we have

TD

(

f

)∩

TD

(

F

)⊆

L

(

f

)

,n

∈Z and(1.7).

Remark 1.6

The general Riccati differential equations

f

=

a

(

z

)+

a

(

z

)

f

+

a

(

z

)

f

can be rewritten as

If

a

,a

,a

are entire functions of finite lower order such that

ρ

(

a

)

<µ

(

a

)and

ρ

(

a

)

<µ

(

a

),then meas(

L

(

f

))≥min{2

π,π/µ

(

a

)}follows from Corollary 2.Clearly,(1.3)is different from our case of Riccati differential equation.

The remainder of this paper is organized as follows:in Section 2,we show some basic properties of Julia limiting directions for entire functions,which contain the relation between transcendental directions and Julia limiting directions.The proof of Theorem 1.3 is given in Section 3,and some examples given here.Our method is somewhat different and simpler than that of[9,17].

2 Basic Property of Julia Limiting Directions

The relation between

TD

(

f

)and

L

(

f

)is important for our proof of Theorem 1.3.Before proving the theorem,we need a result which can be deduced from the proof of[11,Lemma 1]in order to deal with the case that F(

f

)contains an angular domain.

One Friday evening I came home from work to find a big beautiful German shepherd on our doorstep. This wonderful strong animal gave every indication that he intended to enter the house and make it his home. I, however, was wary4. Where did this obviously well-cared-for dog come from? Was it safe to let the children play with a strange dog? Even though he seemed gentle, he still was powerful and commanded respect. The children took an instant liking5 to German and begged me to let him in. I agreed to let him sleep in the basement until the next day, when we could inquire around the neighborhood for his owner. That night I slept peacefully for the first time in many weeks.

Lemma 2.1

Let

f

be analytic in the angular domain

Suppose that

f

(Ω(

z

,θ,δ

))is contained in a simply connected hyperbolic domain in C.Then

for any

δ

∈(0

,δ

).

Now by Lemma 2.1,we establish the relation between transcendental directions and Julia limiting directions as follows:

Lemma 2.2

Let

f

be a transcendental entire function.Then

TD

(

f

)⊆

L

(

f

).

Proof

We first treat the case in which F(

f

)has a multiply connected component.We claim that in this case,

L

(

f

)=[0

,

2

π

).Otherwise,there exists one value

θ

/∈

L

(

f

),so there exist

∊>

0

,a

∈C and arg

a

=

θ

such that

Next,we consider the remaining case that all components of the Fatou set are simply connected.For any given value

θ

∈

TD

(

f

),we assume that

θ

/∈

L

(

f

),so we have Ω(

a,θ,

2

∊

)⊂F(

f

)for two constants

∊>

0 and

a

with arg

a

=

θ

.At the same time,there is an unbounded sequence{

z

}⊂Ω(

a,θ,

2

∊

)such that

as

n

→∞.Clearly,

f

(Ω(

a,θ,∊

))is contained in a simply connected hyperbolic domain.By Lemma 2.1,there exist positive constants

k

and

A

such that

With Lemma 2.2 in hand,for the entire

f

,we can investigate the Julia limiting directions by first finding the transcendental directions.By the radial growth of

e

,that is,|exp(

re

)|=

e

,it is easy to see that

L

(sin

z

)=

L

(cos

z

)=[0

,

2

π

)since

TD

(sin

z

)=

TD

(cos

z

)=[0

,

2

π

).There are a few more examples as follows:

Example 2.3

We recall that Mittag-Leffler function

has the uniform asymptotic behavior[7,Chapter 1,(5.40)]

From this fact,it follows that

Example 2.4

From[19,(6.3.15)],the entire function

in the angle Ω(

ε,

2

π

−

ε

)={

z

:

ε<

arg

z<

2

π

−

ε

}for every positive number

ε

.Since

ε

is arbitrarily small,it is easy to see that

L

(

f

)=

TD

(

f

)=[0

,

2

π

).

Example 2.5

By[19,Lemma 7.9],for 1

/

2

<µ<

1,we know that

is an entire function,and for a sufficiently small

ε>

0,

uniformly in

θ

for|

θ

|

<π

−

ε.

This implies that

Thus,to measure

L

(

f

),one possible way is to estimate the directions in which

f

grows quickly.To do this,we recall Baerstein’s result on the spread relation[1],which shows that for

f

with not so many poles,log|

f

|is‘large enough’on a substantial portion of circles{|

z

|=

r

}.

Lemma 2.6

([1])Let

f

be a transcendental meromorphic function with finite lower order

µ

and positive de ficiency

Clearly,

E

(

f

)⊆

TD

(

f

),so

E

(

f

)⊆

L

(

f

),by Lemma 2.2.Next,by Lemma 2.6 and the monotone convergence theorem,we derive the lower bound of meas(

L

(

f

)).

Lemma 2.8

Let

f

be a transcendental meromorphic function with finite lower order

µ

and

δ

(∞

,f

)

>

0,and let Λ(

r

)be a positive function such that Λ(

r

)=

o

(

T

(

r,f

))and Λ(

r

)

/

(log

r

)→∞as

r

→∞.Then,

Proof It follows from Lemma 2.6 that

Noting that

D

(

r

)⊆

B

for each

n

,we get that

Combining this fact with(2.8)and

E

(

f

)⊆

L

(

f

)yields inequality(2.7).In addition,we easily have the following lemma for

L

(

f

)∩

L

(

f

):

Lemma 2.9

Let

f

be a transcendental entire function,and let

n

be a positive integer.Then

TD

(

f

)⊆

TD

(

f

)and

TD

(

f

)⊆

L

(

f

)∩

L

(

f

).

Proof

For any given

θ

/∈

TD

(

f

),it follows from the de finition of transcendental directions that there exist positive

∊

and

K

such that

We note the fact that

where

c

is a constant,and the integral path is the segment of a straight line from 0 and

z

.From this and(2.9),it is easy to see that|

f

(

z

)|≤(

K

+1)|

z

|for all

z

∈Ω(0

,θ,∊

).Repeating this discussion

n

times yields

This means that

θ

/∈

TD

(

f

)if

θ

/∈

TD

(

f

),which implies that

TD

(

f

)⊆

TD

(

f

).By Lemma 2.2,

TD

(

f

)⊆

L

(

f

)and

TD

(

f

)⊆

L

(

f

),so

TD

(

f

)⊆

L

(

f

)∩

L

(

f

).

3 Proof of Theorem 1.3 and Some Examples

To prove Theorem 1.3,we still need the Nevanlinna theory in angular domains.For the convenience of the reader,we recall some basic de finitions here(for example see[7,22]).

Let

g

(

z

)be an entire function on the closure of Ω(

α,β

)={

z

∈C:arg

z

∈(

α,β

)},where

β

−

α

∈(0

,

2

π

].De fine

where

ω

=

π/

(

β

−

α

)and

b

=|

b

|

e

are the poles of

g

in the closure of Ω(

α,β

)appearing according to their multiplicities.Nevanlinna’s angular characteristic of

g

is de fined by

and the order of

S

(

r,g

)is de fined by

Lemma 3.1

([9,Lemma 2.2])Suppose that

n

∈N,and that

g

(

z

)is analytic in Ω(

α,β

)with

ρ

(

g

)

<

∞.Then,for

ε

=0,

outside a set whose Lebesgue measure is zero,where

and there exist positive constants

M,K

only depending on

g,ε

,ε

,...,ε

,

Ω(

α,β

)such that for all

m

=1

,

2

,...,n

and

z

=

re

∈Ω(

α

,β

),holds outside an R-set,where

k

=

π/

(

β

−

α

)(

j

=1

,

2

,...,n

).

Lemma 3.2

([22,Theorem 2.5.1])Let

f

(

z

)be a meromorphic function on Ω(

α

−

ε,β

+

ε

)for

ε>

0 and 0

<α<β<

2

π

.Then

for

r>

1,possibly excepting a set with finite linear measure,and also we have the constant

K>

0.

Proof of Theorem 1.3

In what follows,we will treat three cases:

n

=0

,n>

0 and

n<

0.Case 1.We assume that

n

=0.For every

θ

/∈

TD

(

f

),by the de finiti on of transcendental direction,there exist positive

∊

and

K

such that

This implies that

ρ

(

f

)

<

∞.By Lemma 3.1,there are positive

M

,K

and

∊

<∊

such that

holds for

z

∈Ω(

θ,∊

)outside an

R

-set

G

,where

H

={

r

=|

z

|

,z

∈

G

}is a set of finite Lebesgue measure,and

m

=1

,

2

,...,k

.

We rewrite(1.6)as

Taking(3.1)and(3.2)into(3.3)yields

for

z

∈Ω(

θ,∊

)outside

G

,where

n

+

n

+

...

+

n

−

s

≥

γ

−

s

≥0.When

µ

(

F

)

>

0,we take Λ(

r

)=

r

with

for a subsequence{

r

}of{

r

}.

For any given

taking(3.6)into(3.4)yields

It follows from the de finition of order by maximum modulus that the above inequality implies that

This is an contradiction,which means that

E

(

F

)

TD

(

f

)=∅

,

so,by Lemma 2.2,

At the same time,

E

(

F

)⊆

TD

(

F

).Therefore,we have

and(1.7)follows from(3.5).

Case 2.We assume that

n>

0.It follows from Lemma 2.9 that

TD

(

F

)∩

TD

(

f

)⊆

TD

(

F

)∩

TD

(

f

),thus

by

E

(

F

)⊆

TD

(

f

),which similarly leads to(1.7).Case 3.We assume that

n<

0.For

θ

/∈

TD

(

f

),we know that

where

∊>

0 and

K

>

0.Thus,

S

(

r,f

)=

O

(1),so

ρ

(

f

)

<

∞.Then,by Lemma 3.1,there are positive

M

,K

and

∊

<∊

such that

holds for

z

∈Ω(

θ,∊

)outside an

R

-set

G

,where

H

={

r

=|

z

|

,z

∈

G

}is of finite Lebesgue measure.It follows from Lemma 3.2 that

with

nε<∊/

2.Repeating this discussion

n

times yields that

At the same time,by Lemmas 2.6 and 2.8,and Remark 2.7,there exists an unbound sequence{

r

}such that all

r

/∈

H

∪

H

,and for

θ

∈

E

(

F

),we have(3.5)and(3.6),where Λ(

r

)is de fined as in Case 1.We rewrite(1.6)as

For

θ

∈

E

(

F

)

TD

(

f

),substituting(3.2),(3.6),(3.7)and(3.8)into the above equation yields

In a fashion similar to Case 1,this is impossible.This means that

E

(

F

)

TD

(

f

)=∅,so,by Lemma 2.2,

Therefore,we have

and(1.7)follows from(3.5)again.

This completes the proof of Theorem 1.3.

Remark 3.3

From the proof of Theorem 1.3,we know that

Finally,we give some examples for applications of Theorem 1.3.

Example 3.4

The solutions of the Mathieu differential equation

Since the set of transcendental directions is closed,we deduce that

Example 3.5

Every non-zero solution of the equation

Example 3.7

Every entire solution of the equation satis fies[0

,

2

π

)=

L

(

f

),since