HUANG Hui-jun,NING Ju-hong

(1.Attached Primary School of Jiangxi Normal University,Nanchang 330027,China;2.College of Mathematics and Information Science,Jiangxi Normal University,Nanchang 330022,China)

Order of Dirichlet Series in the Whole Plane and Remainder Estimation

HUANG Hui-jun1,NING Ju-hong2

(1.Attached Primary School of Jiangxi Normal University,Nanchang 330027,China;2.College of Mathematics and Information Science,Jiangxi Normal University,Nanchang 330022,China)

In this paper,frstly,the ρ order and ρβorder of Dirichlet series which converges in the whole plane are studied.Secondly,the equivalence relation between remainder logarithm lnEn-1(f,α),lnRn(f,α)and coefcients logarithm ln|an|is discussed respectively.Finally, the theory of applying remainder to estimate ρ order and ρβorder can be obtained by using the equivalence relation.

Dirichlet series;ρ order;ρβorder;remainder estimation

§1. Introduction

The equivalence between coefcients|an|and remainder Rn(f,δ)of Dirichlet series which converges in the half-plane are studied in[1].The growth of Dirichlet series which converges in the half-plane is studied by introducing the concept of β-order and type of function U(σ)in [2-4].Besides,the equivalence between remainder En−1(f,α)and coefcients|an|are obtained under certain conditions.

In this paper,the growth of Dirichlet series of infnite order in the whole-plane is discussed and under the growth conditions which are diferent from[1-2],the equivalence relation between remainder logarithm lnEn−1(f,α),lnRn(f,α)and coefcients logarithm ln|an|is discussedrespectively.Finally,using the equivalence relation,we obtain some relations between the error and the order.

Dirichlet series are introduced and it has the form

and

From(1.3)and[5,Lemma 3.1.1],we have

thus

It follows from Valiron formula that the abscissa of uniform convergence of(1.1)are−∞and so the series(1.1)defnes an entire function f(s)in the complex plane.

The maximum modulus and the maximum term of(1.1)can be defned as

M(σ)=M(σ,f)=sup{|f(σ+it)|:t∈R},m(σ)=m(σ,f)=max{|an|eλnσ:n∈N}.

Let D be a set of analytic sum functions defned by(1.1)satisfying(1.2)~(1.4)in Res＜+∞, Dαbe a set of analytic sum functions defned by(1.1)satisfying(1.2)~(1.3)in Res≥α. Obviously,suppose f(s)∈D,−∞＜α＜+∞,then

Defnition 1.1[5]Suppose f(s)∈D,the Ritt order ρ of f(s)represented by(1.1)is defned by

If ρ=0,0＜ρ＜+∞,ρ=+∞,series(1.1)are respectively called zero order,fnite order and infnite order Dirichlet series.

Besides,in this paper,the infnite order Dirichlet series are studied in the whole plane by introducing the concept of β-order[2].

Defnition 1.2Let ℑ denote the class of functions β satisfying the following conditions

(i)β(x)is defned on[a,+∞)(a＞0)and is positive,strictly increasing,diferentiable and tends to+∞as x→+∞;

(ii)When x→+∞,we have xβ0(x)=o(1).

Remark 1.1Let β(x)=logpx,p≥2,then β-order become the p order.

Defnition 1.3Suppose f(s)∈D,β∈ℑ,then the β-order ρβof the f(s)is

§2. ρ Order and ρβOrder of Dirichlet Series in the Whole Plane

Our main conclusions are as follows.

Theorem 2.1[5]Suppose that f(s)∈D,a necessary and sufcient condition that the function f(s)has ρ order is

Lemma 2.1[5,Lemma3.2.1]If Dirichlet series(1.1)satisfes conditions(1.4)and(1.6), then∀ε＞0,∀σ∈R,we have

where k(ε)is a positive constant only depending on ε and f(s).

Lemma 2.2Suppose that f(s)∈D and have β-order ρβ,then

When x＞max{C,a}is sufciently large,by Lagrange mean value theorem,we have

Theorem 2.2Suppose that f(s)∈D and have β-order ρβ,then

Remark 2.1This theorem mainly discussed the infnite order Dirichlet series,so we always suppose that M(σ)→+∞(σ→−∞).

ProofLet

Obviously 0≤η≤∞.In what follows,we frst consider the case of 0＜η＜∞.

For 0＜ε＜η,from(2.1)there exist a positive integer sequence{nk},it follows that

From(2.2)and the relationship between the maximum modulus and the maximum term,it follows that

From the method of gaining σk,we have β(λnk)=−σk(η−ε).Then from(2.4)and the increase of function β(x),we obtain β(lnM(σk))＞β(λnk)=−σk(η−ε).Since ε is arbitrary,

As was proved above:for η=0,we have ρβ≥η;for η=∞,we have ρβ=∞.

Next,we are going to prove that ρβ≤η.The method used here from[7].Firstly,we suppose η＜∞.Then for every ε＞0 from(2.1),when n is sufciently large,we have

From(2.5),we have

For any fxed sufciently small σ＜−1,set I=γ[(1−σ)(η+ε)],where γ(x)is the inverse function of β(x),then β(I)=(1−σ)(η+ε)and we have

If n≤n0and λn≤I,then

(2) If λn＞I,then β(λn)＞β(I),−λnβ(λn)＜−λnβ(I)and by(2.6),we have

From(2.7)~(2.9),we have lnm(σ)≤−σγ[(1−σ)(η+ε)],so β(lnm(σ))≤β[−σγ((1−σ)(η+ε))]and

let h(σ)=γ((1−σ)(η+ε))](σ＜−1),then by the Lagrange mean value theorem,there exists a τ∈(h(σ),−σh(σ)),such that

Due to the arbitrary nature of ε,we get

by Lemma 2.2,we know ρβ≤η.

The above inequality obviously holds,when η=0 and η=∞.

In conclusion,Theorem 2.2 is obtained.

§3. Proof and Application of Equivalence Relation

This section focuses on proving the equivalence relation of remainder logarithm lnEn−1(f,α) and coefcients logarithm ln|an|of Dirichlet series.We give some following basic notations(identical with[1])before the results be described.

For every α(−∞＜α＜+∞)and any fxed positive integer n=1,2,3···,ifletbe a set of all exponential polynomial that such aswhere ak,λk(k= 1,2,···,n)is defned by(1.1),

in which

En(f,α)and Rn(f,α)were also defned in[1].

Lemma 3.1Ifthen for every n≥1,we have

and

Then for arbitrary p∈πn−1(λ)(m＞n),we have

Whence|an|e−αλn≤‖fm−p‖α.In addition,because fm(s)uniformly converges to f(s)in{s: Res≥α},then when m→ ∞,we have‖fm−p‖α→‖f−p‖α.Then by the defnition of En−1(f,α),it follows that

Hence

Theorem 3.1If f(s)∈D,then for arbitrary α(−∞＜α＜+∞),we have

Proof(1)By Lemma 3.1,we can see that|an|≤En−1(f,α)eαλn.Then λn−ln|an|≥λn−ln[En−1(f,α)eαλn].From(1.4),we have

(2)By the defnition of En−1(f,α),when 1≤k≤n,we have

hence

From(3.2),we have λn−ln[En−1(f,α)eαλn]≥λn−ln|an|−lnM(α)−lnm(α)−lnn−lnC.

From(1.4)and above inequalities we have

Since α is arbitrary constant that independent of n,then from(1.4)and(1.6),we obtain

Thus

This completes the proof of the theorem.

Corollary 3.1If f(s)∈D,from(3.1)and(3.2),it follows that

Corollary 3.2If f(s)∈D,from(1.4)and Corollary 3.1,we have

From Theorem 2.1 and Corollary 3.2,it follows that

Theorem 3.2If f(s)∈D,we obtain

From Theorem 2.2 and Theorem 3.1,it follows that

Theorem 3.3If f(s)∈D and it has β-order ρβ,then for arbitrary α(−∞＜α＜+∞), we have

Theorem 3.4If f(s)∈D and it has β-order ρβ,then for arbitrary α(−∞＜α＜+∞), we have

§4.About Remainder Rn(f,α)

On the other hand,if we add the condition

then by(1.3),let h1(0＜h1＜h)be given,there exists a positive integer n1,such that∀n＞n1,

By(4.2),for a sufciently large number M(＞|α|)＞0,there exists a positive integer n2,such that∀n＞n2,

then for all n＞max{n1,n2},we have

From(4.1)and(4.3),we can get

Theorem 4.1If f(s)∈D and

then for arbitrary α(−∞＜α＜+∞),we have

Remark 4.1Through the comparison for Corollary 3.1 and Theorem 4.1,we can see that if En−1(f,α)of(3.3)~(3.6)was substituted for Rn(f,α),these results still hold adding the condition(4.2).

AcknowledgementThe authors are grateful for the anonymous referees for their helpful suggestions and comments which improved the quality of the paper.

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O174.5

A

1002–0462(2015)04–0515–09

Received date:2013-09-22

Foundation item:Supported by the National Natural Science Foundation of China(11171119);Supported by the National Science Foundation of Jiangxi Province(20122BAB211005,2010GQS0103)

Biographies:HUANG Hui-jun(1989-),female,native of Jian,Jiangxi,a teacher of Attached Primary School of Jiangxi Normal University,engages in studying of complex analysis;NING Ju-hong(correspording author)(1977-),female,native of Sanmenxia,Henan,an associate professor of Jiangxi Normal University,engages in studying of complex analysis.

2000 MR Subject Classifcation:30B50