Yang TAN(譚洋)School of Applied Mathematics,Beijing Normal University,Zhuhai 519087,China

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SEVERAL UNIQUENESS THEOREMS OF ALGEBROID FUNCTIONS ON ANNULI?

Yang TAN(譚洋)
School of Applied Mathematics,Beijing Normal University,Zhuhai 519087,China

E-mail:shutongtan@sina.com

AbstractIn this paper,we discuss the uniqueness problem of algebroid functions on annuli,we get several uniqueness theorems of algebroid functions on annuli,which extend the Nevanlinna value distribution theory for algebroid functions on annuli.

Key wordsthe Nevanlinna theory;multiple values;the uniqueness of algebroid functions on annuli

2010 MR Subject Classi fi cation34M10;30D35

?Received September 9,2014;revised June 16,2015.Project Supported by the Natural Science Foundation of China(11171013).

1 Introduction

In 1926,Nevanlinna[1]proved the following famous fi ve-value theorem:

For two nonconstant meromorphic functions f(z)and g(z)on the complex plane C,

if they have the same inverse images(ignoring multiplicities)for fi ve distinct values,

then f(z)≡g(z).

After this wonderful work,the uniqueness theory of meromorphic functions in C attracted many investigations[2-5].As the extension of meromorphic functions,the uniqueness of algebroid functions in the complex plane C is an important subject in the value distribution theory.The uniqueness problem of algebroid functions was firstly considered by Valiron,afterwards some scholars got several uniqueness theorems of algebroid functions in the complex plane C[6-13].In 2005,Khrystiyanyn and Kondratyuk proposed the Nevanlinna theory for meromorphic functions in multiply connected domains[14,15].In 2009,Cao and Yi[16]investigated the uniqueness of meromorphic functions sharing some values and some sets on annuli.Thus it is interesting to consider the uniqueness problem of algebroid functions in multiply connected domains.In this paper,we mainly study doubly connected domain.We assume that the readers are familiar with the Nevanlinna theory of meromorphic functions and algebroid functions[17-27].By the doubly connected mapping theorem[28]each doubly connected domain is conformally equivalent to the annulus A(R1,R2)={z:R1＜|z|＜R2},0≤R1＜R2≤+∞.We only consider two cases:

In the latter case the homothetyreduces the given domain to the annulusThus,in two cases every annulus is invariant with respect to the inversion

2 Basic Notions and De finitions

Let Av(z),Av?1(z),···,A0(z)be a group of holomorphic functions which have no common zeros and de fi ne on the annulus

Then irreducible equation(2.1)de fines a v-valued algebroid function on the annulus(1＜R0≤+∞).

Let W(z)be a v-valued algebroid function on the annulususe the notations:

where wj(z)(j=1,2,···,v)is a one-valued branch of W(z),n1(t,W)is the counting function of poles of the function W(z)in{z:t＜|z|≤1}and n2(t,W)is its counting function of poles in{z:1＜|z|≤t}(both counting multiplicity);is the counting function of poles of the functionis its counting function of poles in{z:1＜|z|≤t}(both ignoring multiplicity);is the countingfunction of poles of the functionwith multiplicity≤k(or＞k)in{z:t＜|z|≤1},each point counts only once;is the counting function of poles of the functionwith multiplicity≤k(or＞k)in{z:1＜|z|≤t},each point counts only once.nx1(t,W)and nx2(t,W)are the counting function of branch points of the function W(z)in {z:t＜|z|≤1}and{z:1＜|z|≤t},respectively.Nx(r,W)is the density index of branch point of W(z)on the annulus

Let W(z)be an algebroid function on the annulusif there are λ branches of W(z)which take a(∞)as the value in z0point,then we have the fractional power series

n0(r,a)=where n0(r,a)is the counting function of zeros of W(z)?a on the annulus(counting multiplicity).If there are λ branches of W(z)which take∞as the value in z0point,then we have the fractional power series

n0(r,∞)=n0(r,W)=where n0(r,∞)is the counting function of poles of W(z)on the annulus(counting multiplicity).z=z0is a branch point of λ?1 degree of W(z)on its Riemann Surfacedenotes the branch points of W(z)on its Riemann Surface on the annulus

Let W(z)be a v-valued algebroid function which be determined by(2.1)on the annulusWhen a =∞,N0(r,W)=are the counting function of zeros of W(z)?a and ψ(z,a)on the annulus

De finition 2.1Let W(z)be an algebroid function on the annulus+∞),the function

is called the Nevanlinna characteristic of W(z).

De finition 2.2Let W(z)be an algebroid function on the annulus+∞),we denote the de fi ciency of a∈C=C∪{∞}by

and denote the reduced de fi ciency by

3 Some Lemmas

Lemma 3.1(see[14])Let f be a nonconstant meromorphic function on the annulus

where 1≤r＜R0.

Lemma 3.2([14],Jensen theorem for meromorphic function on annuli)Let f be a nonconstant meromorphic function on the annulus

where 1≤r＜R0.

Lemma 3.3Let W(z)be a v-valued algebroid function which is determined by(2.1)on the annulus

ProofFirst,we have

So,from above determinant we know that J(z)is a holomorphic function on the annulus.In fact,by(2.2),if there are λ branches of W(z)which take a∈C as the value in z0point,then there areitems including the factorin J(z)(τ is the multiplicity of zero),that is:z0is a zero of J(z),the multiplicity of z0isat least.That is to say,the branch points of λ?1 degree of W(z)are zeros of λ?1 degree of J(z)atleast.So(3.1)is true.By substitutinginto J(z),using Lemma 3.2,we get

So we have

Lemma 3.4(the first fundamental theorem for algebroid function on annuli)Let W(z)be a v-valued algebroid function which is determined by(2.1)on the annulusR0≤+∞),a∈C,

ProofBy Viete theorem,we have

Using Lemma 3.2,we get

Among them

because

So

Lemma 3.5(the second fundamental theorem for algebroid function on annuli)Let W(z)be a v-valued algebroid function which is determined by(2.1)on the annulusR0≤+∞),ak(k=1,2,···,p)are p distinct complex numbers(finite or in finite),then we have

N1(r,W)is the density index of all multiple values including finite or in finite,every τ multiple value counts τ?1,and

ProofLet ak∈C(k=1,2,···,p),wj=wj(z)(j=1,2,···,v)are v branches of W(z),by the following identity

Ck=[(a1?ak)(a2?ak)···(ak?1?ak)(ak+1?ak)···(ap?ak)]?1,w′(z)is the derivative of w(z)and satisfies the following equation

By(3.4),

Among them,

So we have

Let

So we get

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By(3.9),(3.10),(3.11)

Combining(3.6),(3.7),(3.8),(3.12)and using Lemma 3.4 we have

And because

Then

By(3.13)and above formula

Because N0(r,W)≤T0(r,W)+O(1),so(3.14)can be rewritten as the following

So we get(3.16).By substituting(3.16)into(3.15)we have

By(3.17)and Lemma 3.3,we get(3.3).

The remainder of the Second Fundamental Theorem is the following formula,

outside a set of finite linear measure,if r?→R0=+∞;while

outside a set E of r such that

We notice that the following formula is true,

Lemma 3.6(the Cartan theorem for algebroid function on annuli)Let W(z)be a v-valued algebroid function which is determined by(2.1)on the annulusthen we get

ProofLet a be a finite complex number,then we have[2,22]

By(2.1)

We integrate(3.21)on α from 0 to 2π and by(3.20),

By(3.23),(3.24)and(3.25),

By(3.26)and(3.27),(3.19)is true.

Lemma 3.7Let W(z)be a v-valued algebroid function which be determinated by(2.1)on the annulusif the following conditions are satis fied

then W(z)is an algebraic function.

So we have

Because therefore

So we get

On the other hand,there is the following formula by Viete theorem of algebraic equation

where(α1,α2,···,αv?j)is the combination of taking v?j numbers from(1,2,···,v),(?1)αis 1 or-1,which depends onbeing even permutation or odd permutation.Now everyby(3.34),

The right hand side of(3.35)has nothing to do with number j,so any(1＜R0≤+∞)

Then we get

So according to(3.33)and(3.37),we have

According to(3.38)and(3.39),we have

By the conditions of Lemma 3.7 and above formula,all meromorphic functions fjk(z)(0≤j,k≤v)which satisfy the following conditions

By references[14,15]and[22],all functions fjk(z)are rational functions,because A0(z),A1(z),···,Av(z)can’t have nonconstant common factor,so all Aj(z)(j=1,2···v)must be polynomials.Then W(z)degenerates an algebraic function.

Remark 3.8Let W(z)be an algebroid function on the annulus+∞)and let a be a complex number.We useto denote the set of zeros of W(z)?a with multiplicity no greater than k,in which each zero is counted only once.

Remark 3.9Now let W(z)be a v-valued algebroid function which is determined by(2.1)on the annulusbe aμ-valued algebroid function which is determined by the following equation on the annulus

Without loss of generality,letdenotes the counting function of the common values of=a with multiplicity≤k on the annulus+∞),each point counts o︿nly once.And let

4 Main Results

Furthermore let

and

where m and n are positive integers in(1,2,···q)and b is an arbitrary complex number.If

By De finition 2.2

Because

By De finition 2.2 we have

From(4.6)and(4.7)

From(4.4),(4.5)and(4.8)we get

From(4.1)

So we can deduce that

Thus we have

where

By similar discussion we get

where

By(4.9),(4.10)and Remark 3.9

R(?,ψ)denotes the resultant of ?(z,W)and ψ(z,W),it can be written as the following

It can be written in another form

So we know that R(?,ψ)is a holomorphic function,using Lemma 3.2,

Then we get

By the conditions of Theorem 4.1,we know that W(z)andtake the same values with multiplicity≤kjabout q distinct aj,each point counts only once,at the same time we getFrom(4.11),(4.12)and Remark 3.9

Hence

From Lemma 3.7 we know that this is not true.Therefore we complete the proof of Theorem 4.1.

Set

where m and n are positive integers in(1,2,···,q).If

ProofSince δ0(aj,W)≥the assertion follows from Theorem 4.1.

If

where m is positive integer in(1,2,···q),then we have

ProofLetting m=n,Corollary 4.3 immediately follows from Corollary 4.2.

If

ProofLetting m=4v+1,Corollary 4.4 immediately follows from Corollary 4.3.

(ii)If q=8v and kj＞1 then

(iii)If q=7v and kj＞2 then

ProofWe note that

Corollary 4.5 immediately follows from Corollary 4.4.

Thus from Corollary 4.5 we obtain the theorem as following.

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