(孙道椿)

School of Mathematics,South China Normal University,Guangzhou 510631,China E-mail:1457330943@qq.com

Yingying HUO (霍颖盈)

School of Applied Mathematics,Guangdong University of Technology,Guangzhou 510520,China E-mail:huoyingy@gdut.edu.cn

Fujie CHAI (柴富杰)†

School of Financial Mathematics&Statistics,Guangdong University of Finance,Guangzhou 510521,China E-mail:chaifujie@sina.com

Abstract In this paper,we construct k-valued random analytic algebroid functions for the first time.By combining the properties of random series,we study the growth and Borel points of random analytic algebroid functions in the unit disc and obtain some interesting theorems.

Key words Sequence of random variables;algebroid functions;growth;Borel points

1 Introduction and results

In 1948,J.E.Littlewood and A.C.Offord discussed the“pit behaviour”of random integral functions with a Rademacher squence([4]).M.N.Mikhail later obtained some analogous results on random meromorphic functions in the complex plane([5]).In[8],D.C.Sun and J.R.Yu studied some random Dirichlet series with wider assumptions in the half plane and proved that all the points on the imaginary axis are the Borel points almost surely(a.s.).In 2009,D.C.Sun investigated the growth and Borel points of some random meromorphic functions in the unit disc and obtained some interesting theorems(see[9]).In the 1930s,the Nevanlinna value distribution theory for meromorphic functions(see[1,14])was extended to algebroid functions by H.Selberg([6]),E.Ullrich([12])and G.Valiron([13]).The natural question arises:how can we de fine random algebroid functions and discuss their properties of growth and Borel values or points?In this paper,we give the de finition of

k

-valued random algebroid functions for the first time,and prove that for those random algebroid functions,every point on the unit circle is a Borel point a.s..

Unless otherwise speci fied,the reader can refer to[1,2,7,14]for the notation and terminology used in this paper.

Suppose that

A

(

z

)

,A

(

z

)

,...,A

(

z

)are analytic functions in the unit disc{|

z

|

<

1}.Ψ(

z,W

)is a bivariate complex function satisfying

For all

z

∈{|

z

|

<

1},the equation Ψ(

z,W

)has

k

complex roots

w

(

z

)

,w

(

z

),

...,w

(

z

).Then equation(1.1)de fines a

k

-valued algebroid function

W

(

z

).If,for all

z

∈{|

z

|

<

1},we have

A

(

z

)/=0,then

W

(

z

)is an analytic algebroid function,and we can divide both sides of(1.1)by

A

(

z

).Hence,the analytic algebroid function has the form

De finition 1.1

Suppose that

n

(

r

)is a non-negative real valued function in(0

,

1).The order of

n

(

r

)is de fined as

Remark 1.2

Suppose that

W

(

z

)is a

k

-valued algebroid function in the unit disc.The order of

W

(

z

)is de fined as

De finition 1.4

For

z

∈{|

z

|

<R

},we de fine

µ

(

r,A

)as

this is a kind of

N

-sequence in[8].We de fine the

k

random Taylor series

A

(

z,ω

)as

By(1.4),there is at least one

t

∈{1

,

2

,...,k

−1},say

t

=1

,

such that

Hence,the convergent radius of

A

(

z,ω

)is 1 a.s..The convergent radius of

A

(

z,ω

)(

t

=2

,...,k

)is no less than 1 a.s..Then we can de fine the the

k

-valued random analytic algebroid function

W

(

z,ω

)in unit disc as follows:

Furthermore,a random Taylor series(1.3)is a special case of the random Dirichlet series in[8].By Theorem 3 in[8],we can obtain that all points on the circle{

e

|

u

∈[0

,

2

π

)}are the Borel points of order

ρ

+1 of

A

(

z,ω

),with no finite exceptional values a.s..

In this paper,we will prove three theorems.First,we give two theorems on algebroid functions in the unit disc.The main idea of the proofs is from[2,10],in which the properties are proved when algebroid functions are de fined in the whole complex plane,and the proofs of these two theorems are similar to those in the whole complex plane.We give here complete proofs for the convenience of readers and the completeness of this paper.

Theorem 1.5

Let

W

(

z

)be a

k

-valued algebroid function de fined by(1.1)in the disc{|

z

|

<R

}.For

b

∈C,Ψ(

z,b

)/≡0,we have

Theorem 1.6

For a

k

-valued analytic algebroid function

W

(

z

)of order

ρ

(

W

)de fined by(1.2)in unit disc,the equation

holds,where

ρ

(

A

)=max{

ρ

(

A

);

t

=1

,

2

,...,k

}

.

Now,we give the main theorem of this paper.

Theorem 1.7

Let

W

(

z,ω

)be a

k

-valued random analytic algebroid function de fined by(1.5)in the unit disc.For almost surely all

ω

∈Ω,all points on the circle{

e

;

t

∈[0

,

2

π

)}are Borel points of order

ρ

+1.

2 Proof of Theorem 1.5

To prove Theorem 1.5,we need the following lemmas:

Lemma 2.1

Suppose that

f

(

z

),

g

(

z

)are meromorphic functions in{|

z

|

<R

},where

f,g

/≡0 and

f,g

/≡∞.Then for all

r

∈(0

,R

),we have

Proof

For

b

∈C,

a

∈C,let

n

(

z

=

b,f

=

a

)be the multiplicities of

f

(

z

)=

a

at the point

z

=

b

.Suppose that

In the neighbourhood of

b

,we have

When

p

+

q

≥0,we have

When

p

+

q<

0,we have

Therefore,

and denote(

W

+

f

)(

z

)by

W

(

z

)+

f

(

z

).It is easy to see that the function(

W

+

f

)(

z

)=

W

(

z

)+

f

(

z

)is a

k

-valued algebroid function in{|

z

|

<R

}.The de finitional equation of

W

(

z

)+

f

(

z

)is

Lemma 2.3

(Jensen Formula of algebroid function)Suppose that

W

(

z

)is a

k

-valued analytic algebroid function de fined by(1.1)in{|

z

|

<R

}.Then,1.If

A

(

z

)=Ψ(

z,

0)/≡0,then

where

a

is the first non-zero coefficient in the Laurent expansion of the meromorphic function

at

z

=0.2.More generally,if Ψ(

z,b

)/≡0,then

where

c

is the first non-zero coefficient in the Laurent expansion of the meromorphic function

at

z

=

b

.

Proof

By a curve that goes through all branch points,|

z

|

<R

can be divided into a simply connected domain

D

.Let{

w

(

z

)}be

k

single branches of

W

(

z

)in

D

.

i)By the Viete theorem,we have

Using the Jensen Formula of meromorphic functions and Lemma 2.1,we get

where

N

(

r,A

(

z

)=∞)=0=

N

(

r,A

(

z

)=∞).ii)Letting

M

=

W

−

b

,the equation

de fines an algebroid function

M

(

z

).By conclusion(1),and by noticing the fact that Ψ(

z,

0)≡Ψ(

z,b

)≡

B

(

z

),we can obtain conclusion(2).

Proof of Theorem 1.5

It follows from Lemma 2.2 that

Applying the Jensen Formula of algebroid functions in Lemma 2.3,we have

Combining with(2.1),we can obtain Theorem 1.5.

3 Proof of Theorem 1.6

To prove Theorem 1.6,we need several lemmas.

Lemma 3.1

([4])Suppose that

W

(

z

)is a

k

-valued algebroid function in{|

z

|

<R

}.Then

where

c

is the first non-zero coefficient in the Laurent expansion of

A

(

z

)at

z

=0 and

µ

(

r,A

)is de fined in De finition 1.4.

Lemma 3.2

For an algebroid function

W

(

z

)of order

ρ

(

W

)de fined by(1.1)in the unit disc,we have

Proof

By De finition 1.4,

Hence,we can obtain

Applying Lemma 3.1,we have

Remark 3.3

The equal sign in Lemma 3.2 may not hold.For example,let

The 2-valued function

is de fined by the equation

Hence,

ρ

(

W

)=0.However,

ρ

(

A

)=

ρ

(

A

)=

ρ

(

H

)=1

.

Lemma 3.4

Suppose that

W

(

z

)is an analytic algebroid function of order

ρ

(

W

)de fined by(1.2).For

t,u

∈{0

,

1

,

2

,...,k

},we have

where

A

(

z

)/≡0 and

C

is a constant.In particular,if there exist

t,u

∈{0

,

1

,

2

,...,k

}satisfying

Proof

Suppose that

A

(

z

)/≡0.Let

where|

x

|=max{1

,x

}.Then

where

C

is a constant.If there exist

t,u

∈{0

,

1

,

2

,...,k

}satisfying

then according to Lemma 3.1 and(3.1),we can obtain

By applying Lemma 3.2,we have

Proof of Theorem 1.6

Noticing that the

A

(

z

)in expression(1.2)is equal to 1,we have

Therefore,we can easily obtain Theorem 1.6 from Lemma 3.4.

4 Proof of Theorem 1.7

To prove Theorem 1.7,we need a conformal mapping as follows:

Lemma 4.1

For all

∊>

0,mapping

maps the sector{

z

;0

<

|

z

|

<

1}∩{

z

;|arg

z

|

<∊

}to the unit disc{|

w

|

<

1}.There is a constant

b

∈(0

,

1)depending on

∊

,such that

Proof

It follows from

Let

z

=

pe

=

p

cos

φ

+i

p

sin

φ

,where

p

∈(

b,

1)and|

φ

|≤

∊

.Let

By(4.1),we have

It follows from(4.2),(4.3)and(4.4)that

Lemma 4.2

Suppose that

n

(

r

)is a positive real valued function in(0

,

1).If

Proof

This Lemma has been proved in[11].For the convenience of the reader,we give here a brief proof.Suppose that

p

(

n

(

r

))=

p

≥1,so for any

∊>

0,we have

where

C>

0 is a sufficiently large constant.Since

we have

p

(

n

(

r

))≤

p

(

N

(

r

))+1

.

Proof of Theorem 1.7

According to[8],we know that all points on circle{

e

;

t

∈[0

,

2

π

)}are the Borel points of order

ρ

+1 of

A

(

z,ω

),with no finite exceptional values a.s.,that is,for all

ω

∈Ω−

E

,

t

∈[0

,

2

π

),

a

∈

C

,

δ

∈(0

,π

),where

P

(

E

)=0,we have

Mapping(4.1)maps sector

D

:={|

z

|

<

1}∩{|arg

z

|

<∊

}to unit disc

B

:={|

w

|

<

1}.Its inverse mapping is denoted by

z

(

w

).By Lemma 3.4,we have

that is,the order of

n

(

r,A

(

z

(

w

)

e

,ω

)=

b

)is no less than

ρ

+1.Hence,we know from Lemma 4.2 that

By Theorem 1.6,we have

For any non-exceptional value of

W

(

z

(

w

)

e

,ω

),say

d

∈C,we have

Applying Lemma 3.2,we have

Then we have

for the arbitrary value of

d

,and

e

is the Borel point of

W

(

z,ω

)of order more than

ρ

+1.Since

e

is a Borel point of order

ρ

+1.