On the Completeness of the System {𝑧^𝜆𝑛log^𝑗𝑧} in 𝐿^2𝑎

Abstract

Let 𝐿^2𝑎(Ω) denote a Hilbert space consisting of analytic functions on an unbounded domain Ω located outside an angle domain with vertex at the origin. We obtain a completeness theorem for the system 𝑀_Λ= {𝑧^𝜆𝑛log^𝑗𝑧,𝑗=0,1,…,𝑚_𝑛−1}^∞𝑛=1, in 𝐿^2𝑎(Ω).

1. Introduction

Let Ω denote a domain in the complex 𝑧 plane. Let 𝐿^2𝑎(Ω) denote the space consisting of all functions 𝑓 analytic in Ω with _Ω||𝑓(𝑧)||²d𝑚<∞,𝑧=𝑥+𝑖𝑦,(1.1) where d𝑚 is the area element in the 𝑧 plane (i.e., d𝑚=d𝑥d𝑦=𝑟d𝑟d𝜃 for 𝑧=𝑟𝑒^𝑖𝜃). It is well known that, with the inner product ⟨𝑓,𝑔⟩=_Ω𝑓(𝑧)𝑔(𝑧)d𝑚,(1.2) and the norm ‖𝑓‖=⟨𝑓,𝑓⟩^1/2, 𝐿^2𝑎(Ω) is a Hilbert space (see, e.g., [1, Chapter  1]).

We say a system {ℎ_𝑛}⊂𝐿^2𝑎(Ω) is complete in 𝐿^2𝑎(Ω) if its linear span is dense in 𝐿^2𝑎(Ω) (see, e.g., [2–7]). The following lemma provides an elementary fact on completeness (see, e.g., [1, Chapter   1] and [3, Lemma  1.1]).

Lemma 1.1. A necessary and sufficient condition for the system {ℎ_𝑛} to be complete in 𝐿^2𝑎(Ω) is that for any 𝑓∈𝐿^2𝑎(Ω), if ⟨𝑓,ℎ_𝑛⟩=0 for all ℎ_𝑛, then 𝑓(𝑧)≡0.

A Dzhrbasian domain (see, e.g., [8]) is an unbounded simply connected domain satisfying the following conditions:

Condition Ω(I). For 𝑟>0, let 𝜎(𝑟) denote the linear measure of the intersection of the circle |𝑧|=𝑟 and Ω. There exists 𝑟₀>0 such that, for 𝑟>𝑟₀, 𝜎(𝑟)≤𝑒^{−𝛼(𝑟)},(1.3) where 𝛼(𝑟)>0 satisfies𝛼(𝑟)=𝛼𝑟₀+^𝑟𝑟₀𝜑(𝑡)𝑡d𝑡(1.4) with 𝜑(𝑟)≥0 and 𝜑(𝑟)↑∞ as 𝑟→∞.

Condition Ω(II). The complement of Ω consists of 𝑚 unbounded simply connected domains Ω_𝑗(𝑗=1,2,…,𝑚), each containing an angle domain Δ_𝑗 with opening 𝜋/𝛽_𝑗,𝛽_𝑗>1/2.

For a Dzhrbasian domain satisfying Condition Ω(I) and Condition Ω(II), Dzhrbasian proved that if ^∞𝛼(𝑟)𝑟^1+𝜗d𝑟=+∞(1.5) (here ∫^∞ means that the lower limit of the integral is sufficiently large), where𝜗=max𝛽₁,…,𝛽_𝑚,(1.6) then the polynomial system {1,𝑧,𝑧²,…} is complete in 𝐿^2𝑎(Ω).

Motivated by the result of Dzhrbasian, Shen [4–7] studied the completeness of the system {𝑧^𝜆𝑛} in 𝐿^2𝑎(Ω), where {𝜆_𝑛} is a sequence of complex numbers satisfyingthe𝜆_𝑛arealldistinctandlim_𝑛→∞||𝜆_𝑛||=∞,(1.7)lim_𝑛→∞𝑛||𝜆_𝑛||=𝐷,0<𝐷<∞,(1.8)Re𝜆_𝑛>0,||Im𝜆_𝑛||≤𝐴(1.9) for some constant 𝐴. Shen also supposed that Ω is a Dzhrbasian domain with the vertex of Δ₁ at the origin (hence 0 is outside of Ω). The following result was obtained in [4, 5].

Theorem A. Assume that the sequence {𝜆_𝑛} satisfy (1.7), (1.8), and (1.9), and Ω is a Dzhrbasian domain which satisfies Ω(𝐼), Ω(𝐼𝐼). If 2𝛽₁(1−𝐷)<1,(1.10) and, for some 𝜀₀, ^∞𝛼(𝑟)𝑟^1+𝜂d𝑟=∞,(1.11) where 𝜂=max𝜗,11/𝛽₁−2(1−𝐷)+𝜀⁰,(1.12) then the system {𝑧^𝜆𝑛} is complete in 𝐿^2𝑎(Ω).

An improved version of Shen's result is given in [3]. Let Ω be a Dzhrbasian domain with the added requirement ofΔ₁=𝑧∶||arg(𝑧)−𝜋||<𝜋2𝜛,(1.13) where 𝜛>1/2 is some constant. In [3], for such a domain, results on the completeness on {𝑧^𝜆𝑛} in 𝐿^2𝑎(Ω) were obtained, assuming {𝑧^𝜆𝑛} is a sequence of complex numbers satisfying (1.7) and (1.8), but (1.9) is replaced by the more general condition||arg𝜆_𝑛||<𝛽<𝜋2,(1.14) thus allowing ℑ(𝜆_𝑛)→0 as 𝑛→∞.

More accurately, the main result in [3] is described as follows.

Theorem B. Assume that the sequence {𝜆_𝑛} satisfy (1.7), (1.8), and (1.14), and Ω is a Dzhrbasian domain which satisfies Ω(𝐼), Ω(𝐼𝐼), and (1.13). Moreover, assume that 2𝜛(1−𝐷cos𝛽)<1.(1.15) Let 𝜂=max𝜗,1ℎ+𝜀⁰,(1.16) where 𝜗 is defined in (1.6), 𝜀₀ is some positive number, and ℎ=max_{0<𝛿<𝐷cos𝛽−1+1/2𝜛}2𝛿√𝐷²sin²𝛽+𝛿²𝐷cos𝛽−1+21𝜛−𝛿.(1.17) If ^∞𝛼(𝑟)𝑟^1+𝜂d𝑟=+∞,(1.18) then the system {𝑧^𝜆𝑛} is complete in 𝐿^2𝑎(Ω).

Remark 1.2. The ℎ in (1.17) is well defined, for reference we refer to [3, Remark  4].

In this paper, motivated by the work in [3–7], we will investigate the completeness of the system 𝑀_Λ={𝑧^𝜆𝑛log^𝑗𝑧,𝑗=0,1,…,𝑚_𝑛−1}^∞𝑛=1 in 𝐿^2𝑎(Ω), where Ω is a Dzhrbasian domain with the added requirement of Δ₁ satisfying (1.13). The system 𝑀_Λ is associated with the multiplicity sequence Λ={𝜆_𝑛,𝑚_𝑛}^∞𝑛=1, that is, a sequence where {𝜆_𝑛} are complex numbers with 𝜆_𝑛≠𝜆_𝑚 wherever 𝑛≠𝑚, and each 𝜆_𝑛 having multiplicity equal to 𝑚_𝑛. The sequence Λ satisfies (1.7), (1.14) and alsolim𝑛^Λ(𝑡)𝑡=𝐷,0<𝐷<∞,(1.19) where 𝑛_Λ(𝑡)=∑_{|𝜆𝑛|≤𝑡}𝑚_𝑛 is the so-called counting function of the sequence Λ. We note that when 𝑚_𝑛=1 for all 𝜆_𝑛, the above relation is equivalent to (1.8). To describe even further the sequence Λ, thus the system 𝑀_Λ as well, we need some definitions from [9]. We denote by 𝐋(𝐜,𝐃) the class of all complex sequence 𝐀={𝑎_𝑛}, |𝑎_𝑛|≤|𝑎_𝑛+1| satisfying the following properties: (1) 𝑛/|𝑎_𝑛|→𝐷≥0,(2) for 𝑛≠𝑘 one has that |𝑎_𝑛−𝑎_𝑘|≥𝑐|𝑛−𝑘| for some constant 𝑐 and (3) sup|arg(𝑎_𝑛)|<𝜋/2. The following definition is from [9].

Definition 1.3. Let the sequence 𝐀∈𝐋(𝐜,𝐃) and 𝑎, 𝑏 be real positive numbers such that 𝑎+𝑏<1. We say that a sequence 𝐁={𝑏_𝑛}^∞𝑛=1 belongs to the class 𝐀_𝑎,𝑏 if for all n ∈ℕ we have 𝑏_𝑛∈𝑧∶||𝑧−𝑎_𝑛||≤𝑎^𝑎𝑛,(1.20) and for all 𝑘≠𝑛 one of the following holds: (i)𝑏_𝑘=𝑏_𝑛, (ii)|𝑏_𝑘−𝑏_𝑛|≥max{𝑒^{−|𝑎𝑘|𝑏},𝑒^{−|𝑎𝑛|𝑏}}.
We may write B in the form of a multiplicity sequence Λ={𝜆_𝑛,𝑚_𝑛}^∞𝑛=1, by grouping together all those terms that have the same modulus and ordering them so that |𝜆_𝑛|<|𝜆_𝑛+1|. This form of B is called as {𝜆,𝑚}  reordering (see [9]).
We prove the elementary fact.

Lemma 1.4. Suppose Ω is a Dzhrbasian domain such that Conditions Ω (I), Condition Ω (II), and (1.13) are satisfied. Moreover, suppose Λ={𝜆_𝑛,𝑚_𝑛}^∞𝑛=1 is a sequence of complex numbers which is a {𝜆,𝑚} reordering of 𝐁={𝑏_𝑛}∈𝐀_𝑎,𝑏 of a sequence 𝐀={𝑎_𝑛}∈𝐋(𝐜,𝐃) such that arg(𝑎_𝑛)→0 as 𝑛→∞, satisfying (1.14). Then 𝑧^𝜆𝑛log^𝑚𝑛𝑧∈𝐿^2𝑎(Ω).

Proof. Due to the definition of the domain Δ₁, the principal branch of log𝑧, that is, log𝑧, is well defined on Ω. Thus, (log^𝑗𝑧)(𝑧^𝜆𝑛)=log^𝑗𝑧exp{𝜆_𝑛log𝑧} is an analytic function in Ω. Let 𝑧=𝑟𝑒^𝑖𝜃 and 𝜆_𝑛=|𝜆_𝑛|𝑒^𝑖𝜃𝑛. Considering |𝜃_𝑛|<𝛽<𝜋/2 and |𝜃|<𝜋−𝜋/2𝜛 whenever 𝑧∈Ω, there exists some positive constant 𝐴 such that for 𝑧∈Ω. ||𝑧^𝜆𝑛log^𝑚𝑛𝑧||<(𝐴𝑟)^{|𝜆𝑛|+𝑚𝑛}.(1.21) Since 𝜑(𝑟)↑∞ as 𝑟→∞, for 𝑟 sufficiently large which is denoted by 𝑟₁, we have 𝜑(𝑟)>2(|𝜆_𝑛|+𝑚_𝑛)+2. Without loss of generality, we can suppose 𝑟₁>𝑟₀. By (1.4), we have 𝑒^{−𝛼(𝑟)}𝑟^{2(|𝜆𝑛|+𝑚𝑛)}<𝑒𝑥𝑝−2||𝜆_𝑛||+𝑚_𝑛+2𝑟^{2(|𝜆𝑛|+𝑚𝑛)}=𝑟^{2(|𝜆𝑛|+𝑚𝑛)1}𝑟²⁴.(1.22) Thus, we have _Ω||𝑧^𝜆𝑛log^𝑚𝑛𝑧||²d𝑥d𝑦≤≤2𝜋^𝑟0₀𝐴2𝜋𝑟(𝐴𝑟)^{2(|𝜆𝑛|+𝑚𝑛)}_{2(|𝜆𝑛|+𝑚𝑛)}d𝑟+^∞𝑟₀𝜎(𝑟)(𝐴𝑟)^{2(|𝜆𝑛|+𝑚𝑛)}d𝑟𝑟^{2(|𝜆𝑛|+𝑚𝑛0)+2}2||𝜆_𝑛||+𝑚_𝑛+2+^∞𝑟₀<∞.𝑒^{−𝛼(𝑟)}(𝐴𝑟)^{2(|𝜆𝑛|+𝑚𝑛)}d𝑟(1.23)

The main result of this paper is as follows.

Theorem 1.5. Suppose that Ω is a Dzhrbasian domain such that Conditions Ω (I), Ω (II), and (1.13) are satisfied. Moreover, suppose that Λ={𝜆_𝑛,𝑚_𝑛}^∞𝑛=1 is a sequence of complex numbers which is a {𝜆,𝑚} reordering of 𝐁={𝑏_𝑛}∈𝐀_𝑎,𝑏 of a sequence 𝐀={𝑎_𝑛}∈𝐋(𝐜,𝐃) such that arg(𝑎_𝑛)→0 as 𝑛→∞, satisfying (1.14). If ^∞𝛼(𝑟)𝑟^1+𝜂d𝑟=+∞,(1.24) where 𝜂 is defined in (1.16), 𝜗 is defined in (1.6), and ℎ is defined in (1.17), then the system 𝑀_Λ is complete in 𝐿^2𝑎(Ω).

The paper is organized as follows. In Section 2, crucial lemmas in proving Theorem 1.5 will be presented. In Section 3, the completeness theorem above will be proved.

2. Preliminary Lemmas

We consider the function𝐺(𝑧)=^∞_𝑛=11−𝜆𝑧^22𝑛^𝜇𝑛,(2.1) where 𝜇_𝑛 denotes the multiplicity of the term 1−𝑧²/𝜆^2𝑛 and the integral𝐾(𝑠)=−12𝜋𝑖^+∞−∞𝑒^{−𝑖𝑦𝑠}𝐺(𝑖𝑦)d𝑦,𝑠=𝑢+𝑖𝑣.(2.2)

For sufficiently small 𝛿>0, denote𝐵_𝛿={𝑠=𝑢+𝑖𝑣∶|𝑣|≤𝜋𝐷cos𝛽−𝛿𝜋}.(2.3) Under the Conditions Ω (I), Ω (II), and Definition 1.3, by [10], we can get the following estimates which will play an important role in the proof of Theorem 1.5.

Lemma 2.1. Given 𝜀>0, 1||𝐾(𝑖𝑦)||≤𝐴(𝜀)𝑒^{(−𝜋𝐷cos𝛼+𝜀)|𝑦|},(2.4) where A(𝜀) is a constant which depends only on 𝜀.

Lemma 2.2. There exists a sequence {𝑡_𝑘} with 𝑘≥𝑡_𝑘≥(1−𝜆)𝑘 (𝜆 is some sufficiently small positive number) such that, for 𝑠=𝑢+𝑖𝑣∈𝐵_𝛿,Re𝑠=𝑢≥0, |||||𝐾(𝑠)−_{||𝜆𝑛||<𝑡𝑘𝑚𝑛−1𝑚}₌₀𝑎_𝑛,𝑚𝑠^𝑚𝑒^{−𝜆𝑛𝑠}|||||≤𝐴_𝑡𝑘𝑒^{−𝑢𝑡𝑘sin(𝜇𝜋)},(2.5) and, for 𝑠=𝑢+𝑖𝑣∈𝐵_𝛿,Re𝑠=𝑢≤0, |||||𝐾(𝑠)−_{||𝜆𝑛||<𝑡𝑘𝑚𝑛−1𝑚}₌₀𝑎_𝑛,𝑚𝑠^𝑚𝑒^{−𝜆𝑛𝑠}|||||≤𝐴_𝑡𝑘𝑒^𝑢𝑡𝑘,(2.6) where A is a constant independent of 𝑠 and 𝑡_𝑘, while 𝜇 is a small positive number satisfying tan(𝜇𝜋)<𝛿𝐷sin𝛽.(2.7)

Let 𝑧=𝑒^𝜉,𝜉=𝜉₁+𝑖𝜉₂ and denote the image of Ω in the 𝜉 plane by Ω^. It follows from Condition Ω(II) and (1.13) that Ω^ must be located inside the strip 𝐵_𝜉=𝜉=𝜉₁+𝑖𝜉₂∶||𝜉₂||<𝜋1−12𝜛.(2.8) Denote 𝐵_𝜛=𝑠=𝑢+𝑖𝑣∶|𝑣|<𝜋𝐷cos𝛽−𝜋1−1𝐵2𝜛_𝛿𝜛,=𝑠=𝑢+𝑖𝑣∶|𝑣|≤𝜋𝐷cos𝛽−𝛿𝜋−𝜋1−12𝜛.(2.9) Suppose that 2𝜛(1−𝐷cos𝛽)<1(2.10) from which 𝜋𝐷cos𝛽−𝜋(1−1/2𝜛)>0 follows, choosing 𝛿 sufficiently small so that0<𝛿<𝐷cos𝛽−1+12𝜛.(2.11) It is obvious that if 𝑠∈𝐵^𝛿𝜛 and 𝜉∈Ω′, then |Im(𝑠−𝜉)|<𝜋𝐷cos𝛽−𝛿𝜋, that is, 𝑠−𝜉∈𝐵_𝛿 in (2.3). Thus for any 𝑓(𝑧)∈𝐿^2𝑎(Ω), we can define a function for 𝑠∈𝐵^𝛿𝜛 by𝐹(𝑠)=_Ω′𝑓𝑒^𝜉||𝑒^𝜉||²𝐾(𝑠−𝜉)d𝜉₁d𝜉₂,𝜉=𝜉₁+𝑖𝜉₂.(2.12)

Remark 2.3. By Lemma  2.6 in [10], when 𝜉∈Ω^ is fixed 𝐾(𝑠−𝜉) is analytic for 𝑠∈𝐵^𝛿𝜛; when 𝑠∈𝐵^𝛿𝜛 is fixed, 𝐾(𝑠−𝜉) is both measurable and bounded for 𝜉∈Ω^. Thus, it is not hard to prove that 𝐹(𝑠) is analytic and bounded in 𝐵^𝛿𝜛 (see [11, Chapter 10, Exercise 16; 1, Section 3] and [3, page 8]).

The following lemma will be crucial in our proof of Theorem 1.5.

Lemma 2.4. If for 𝑠∈𝐵^𝛿𝜛, 𝐹(𝑠)≡0 where 𝐹(𝑠) is defined by (2.12), then _Ω𝑓(𝑧)𝑧^𝑛d𝑥d𝑦=0,𝑛=0,1,2,….(2.13)

Proof. See [3, Lemma 2.4].

We end this section by presenting two more lemmas. The first one is the so-called Carleman's Theorem (see [12, page 103]).

Lemma 2.5. Let log⁻𝑟=max{−log𝑟,0}. If 𝑔(𝑤) is analytic and bounded in the half-plane Im(𝑤)≥0 and ^+∞−∞log⁻||𝑔(𝑡)||1+𝑡²d𝑡=∞,(2.14) then 𝑔(𝑤)≡0.

We also need a result of M. M. Dzhrbasian (see [13, Section 10, Lemma  1]).

Lemma 2.6. Suppose 𝛼(𝑟) be given as in (1.4), let 𝑀Φ_𝑛(𝑟=)=s^∞𝑟₀u𝑒p^{−𝛼(𝑟)}_𝑛≥1𝑟^𝑛d𝑟,𝑟^𝑛√𝑀_2𝑛.(2.15) Then there exists some constant 𝐴>0 such that for 𝑟 sufficiently large logΦ(𝑟)≥𝐴𝛼(𝑟).(2.16)

3. Proof of Theorem 1.5

Proof. Let us fix some notations. Throughout this section, 𝐴 will denote positive constants, and it may be different at each occurrence.
To prove Theorem 1.5, it suffices to show that if 𝑓∈𝐿^2𝑎(Ω) and 𝑓(𝑧),𝑧^𝜆𝑛log^𝑗𝑧=0,𝑗=0,1,2,…,𝑚_𝑛−1,𝑛=1,2,…,(3.1) then 𝑓(𝑧)≡0. We claim that, by letting 𝐹(𝑠) be the function as in (2.12), we only need to prove 𝐹(𝑠)≡0 for 𝑠∈𝐵^𝛿𝜛. Indeed by Lemma 2.4, it follows that (2.13) is satisfied, that is ⟨𝑓(𝑧),𝑧^𝑛⟩=0,𝑛=0,1,…. Since (1.24) holds, by Dzhrbasian's result the system {𝑧^𝑛} is complete in 𝐿^2𝑎(Ω) which means 𝑓(𝑧)≡0. Our claim is now justified.
For 𝑠∈𝐵_𝛿, let {𝑡_𝑘} be the sequence defined in Lemma 2.2, with 𝑘≥𝑡_𝑘≥(1−𝜆)𝑘 where 𝜆 is a sufficiently small positive number. Then 𝐹(𝑠)=_Ω′𝑓𝑒^𝜉||𝑒^𝜉||²𝐾(𝑠−𝜉)d𝜉₁d𝜉₂,𝜉=𝜉₁+𝑖𝜉₂=_Ω′𝑓𝑒^𝜉||𝑒^𝜉||²⎡⎢⎢⎣𝐾(𝑠−𝜉)−_{||𝜆𝑛||<𝑡𝑘}^𝑚𝑛−1_𝑚₌₀𝑎_𝑛,𝑚(𝑠−𝜉)^𝑚𝑒^{−𝜆𝑛(𝑠−𝜉)}⎤⎥⎥⎦d𝜉₁d𝜉₂+_Ω′𝑓𝑒^𝜉||𝑒^𝜉||²⎛⎜⎜⎝_{||𝜆𝑛||<𝑡𝑘}^𝑚𝑛−1_𝑚₌₀𝑎_𝑛,𝑚(𝑠−𝜉)^𝑚𝑒^{−𝜆𝑛(𝑠−𝜉)}⎞⎟⎟⎠d𝜉₁d𝜉₂=∶𝐹_1,𝑘(𝑠)+𝐹_2,𝑘(𝑠).(3.2) Since 𝑓(𝑧),𝑧^𝜆𝑛log^𝑗𝑧=0,𝑗=0,1,2,…,𝑚_𝑛−1,𝑛=1,2,…,(3.3) we have 𝐹_2,𝑘(𝑠)=0.(3.4) Hence, for 𝑠=𝑢+𝑖𝑣∈𝐵_𝛿, 𝐹(𝑠)=𝐹_1,𝑘(𝑠). By (2.5) and (2.6) in Lemma 2.2, we have ||𝐹(𝑠)||=≤||𝐴𝐹_{1,𝑘𝑡𝑘}(𝑠)||𝑒^{−𝑢𝑡𝑘sin(𝜇𝜋)}_{Ω′⋂{Re(𝑠−𝜉)≥0}}|||𝑓𝑒^𝜉|||||𝑒^𝜉||²||𝑒^𝜉||^{𝑡𝑘sin(𝜇𝜋)}d𝜉₁d𝜉₂+𝑒^{−𝑢𝑡𝑘}_{Ω′⋂{Re(𝑠−𝜉)≤0}}|||𝑓𝑒^𝜉|||||𝑒^𝜉||²||𝑒^𝜉||^𝑡𝑘d𝜉₁d𝜉₂,(3.5) where 𝐴 is a constant independent of 𝑘 and 𝑠. Hence, for Re𝑠=𝑢≥0, ||𝐹(𝑠)||≤𝐴^𝑡𝑘∬^Ω||𝑓(𝑧)|||𝑧|^𝑡𝑘d𝑥d𝑦|𝑒^𝑠|^{𝑡𝑘sin(𝜇𝜋)}+∬^Ω||𝑓(𝑧)|||𝑧|^𝑡𝑘d𝑥d𝑦≤𝐴_𝑡𝑘|𝑒^𝑠|^𝑡𝑘∬_Ω||𝑓(𝑧)|||𝑧|^𝑡𝑘d𝑥d𝑦|𝑒^𝑠|^{𝑡𝑘sin(𝜇𝜋)}.(3.6) By Schwarz'z inequality ||𝐹(𝑠)||≤𝐴^𝑡𝑘∬^Ω||𝑓(𝑧)||²d𝑥d𝑦^1/2∬_Ω|𝑧|^2𝑡𝑘d𝑥d𝑦^1/2|𝑒^𝑠|^{𝑡𝑘sin(𝜇𝜋)},(3.7) and, by Condition Ω(I), we have the estimate _Ω|𝑧|^2𝑡𝑘d𝑥d𝑦≤𝐴^𝑡𝑘^∞𝑟₀𝑒^{−𝛼(𝑟)}𝑟^2𝑡𝑘d𝑟,(3.8) where 𝐴 is some positive constant independent of 𝑘 and 𝑠. Thus, by 𝑘≥𝑡_𝑘≥(1−𝜆)𝑘, we have ||𝐹(𝑠)||≤𝐴^𝑡𝑘∫^∞𝑟₀𝑒^{−𝛼(𝑟)}𝑟^2𝑡𝑘d𝑟^1/2|𝑒^𝑠|^{𝑡𝑘sin(𝜇𝜋)}≤𝐴^𝑘∫^∞𝑟₀𝑒^{−𝛼(𝑟)}𝑟^2𝑘d𝑟^1/2|𝑒^𝑠|^{(1−𝜆)𝑘sin(𝜇𝜋)},(3.9) for every 𝑘=1,2,…. Hence, ||𝐹(𝑠)||≤inf_𝑘≥1⎧⎪⎨⎪⎩𝐴_𝑘∫^∞𝑟₀𝑒^{−𝛼(𝑟)}𝑟^2𝑘d𝑟^1/2|𝑒^𝑠|^{(1−𝜆)𝑘sin(𝜇𝜋)}=inf_𝑘≥1⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩∫^∞𝑟₀𝑒^{−𝛼(𝑟)}𝑟^2𝑘d𝑟^1/2(1/𝐴)|𝑒^𝑠|^{(1−𝜆)sin(𝜇𝜋)}^𝑘⎫⎪⎬⎪⎭.(3.10)
Let 𝑡=1𝐴||𝑒^𝑠𝑀_𝑛||^{(1−𝜆)sin(𝜇𝜋)},=^∞𝑟₀𝑒^{−𝛼(𝑟)}𝑟^𝑛d𝑟.(3.11) Then ||𝐹(𝑠)||≤inf_𝑛≥1√𝑀_2𝑛𝑡^𝑛.(3.12) If we let Φ(𝑡)=sup_𝑛≥1𝑡^𝑛√𝑀_2𝑛,(3.13) then it follows from Lemma 2.6 that there is some constant 𝑞>0 so that Φ(𝑡)>𝑒^{𝑞𝛼(𝑡)}.(3.14) Combining (3.12) and (3.14) shows that for ℜ𝑠≥0||𝐹(𝑠)||≤𝑒^{−𝐴𝛼(𝑟)},𝑟=1𝐴||𝑒^𝑠||^{(1−𝜆)sin(𝜇𝜋)}.(3.15) In order to use Lemma 2.5, we transform the domain 𝐵_𝛿 into the upper half-plane Im𝑧≥0. (i)First, let 𝑧₁=𝑒^𝑠, 𝐵_𝛿 is then transformed into an angle |arg𝑧₁|≤𝑚𝜋, where 𝑚=𝐷cos𝛽−𝛿−1+12𝜛.(3.16)(ii)Let 𝑧₂=𝑧^11/2𝑚. The above angle domain is transformed into the right half-plane Re𝑧₂≥0. (iii)Finally, let 𝑧=𝑖𝑧₂; the right half-plane is then transformed into the upper half-plane Im𝑧≥0.
More accurately, we have 𝐹(𝑠)=||𝐹𝑒^𝑠||l=og||𝑧𝑧₁₁||=||𝑧^22𝑚||=||(−𝑖𝑧)^2𝑚||=||𝑧^2𝑚||,=𝐹log𝑧^22𝑚=𝐹log(−𝑖𝑧)^2𝑚.(3.17)
Define 𝑔(𝑧)=𝐹(log(−𝑖𝑧)^2𝑚); it is obvious that 𝑔(𝑧) is analytic and bounded in the upper half-plane Im𝑧≥0. By (3.15), for Im𝑧≥0 and |𝑧| sufficiently large, we have ||𝑔(𝑧)||≤𝑒^{−𝐴𝛼(𝐴|𝑧|2𝑚(1−𝜆)sin(𝜇𝜋))}=𝑒^{−𝐴𝛼(𝐴|𝑧|𝑚′)},(3.18) where 𝐴 is some positive constant independent of 𝑧, 𝑚 is given by (3.16), and 𝑚^=2𝑚(1−𝜆)sin(𝜇𝜋)=2𝐷cos𝛽−𝛿−1+12𝜛(1−𝜆)sin(𝜇𝜋).(3.19) Let tan(𝜇𝜋)→𝛿/(𝐷sin𝛽) in (2.7), then sin(𝜇𝜋)⟶𝛿√𝐷²sin²𝛽+𝛿².(3.20)
Denote 𝑚^=2𝛿√𝐷²sin²𝛽+𝛿²𝐷cos𝛽−𝛿−1+21𝜛(1−𝜆).(3.21) By (3.18), for Im𝑧≥0 and |𝑧| sufficiently large, we have ||𝑔(𝑧)||≤𝑒^{−𝐴2𝛼(𝐴3|𝑧|𝑚′′)}.(3.22) It is obvious that 𝛿 can be chosen such that 0<𝛿<𝐷cos𝛽−1+1/2𝜛.
Denote ℎ^=max_{0<𝛿<𝐷cos𝛽−1+1/2𝜛}𝑚^.(3.23) By (3.22), for Im𝑧≥0 and |𝑧| sufficiently large, we have ||𝑔(𝑧)||≤𝑒^{−𝐴2𝛼(𝐴3|𝑧|ℎ′)}.(3.24) Since ℎ^=ℎ(1−𝜆), choosing 𝜆 sufficiently small yields 1ℎ^<1ℎ+𝜀⁰,(3.25) where 𝜀₀ is defined in (1.16). Thus, by (3.24), ^∞log||𝑔(𝑡)||𝑡²d𝑡≤−𝐴^∞𝛼(𝑤)𝑤^1+1/ℎ′d𝑤,(3.26) where 𝐴 is some positive constant independent of 𝑤=𝑐𝑡^ℎ. Thus, by (1.24), we have ^∞log||𝑔(𝑡)||𝑡²d𝑡=−∞.(3.27) Hence ^∞log||𝑔(𝑡)||1+𝑡²d𝑡=−∞.(3.28) Let ∫_−∞ mean that the upper limit of the integral is a negative number with sufficiently large magnitude. Similarly, we have _−∞log||𝑔(𝑡)||𝑡²d𝑡≤_−∞−𝐴₂𝛼𝐴₃|𝑡|^ℎ′𝑡²d𝑡=^∞−𝐴₂𝛼𝐴₃𝑡^ℎ′𝑡²d𝑡=−∞.(3.29) Hence _−∞log||𝑔(𝑡)||1+𝑡²d𝑡=−∞.(3.30) By Remark 2.3, we know that ^𝑏𝑎log||𝑔(𝑡)||1+𝑡²d𝑡<+∞(3.31) for every finite closed interval [𝑎,𝑏], thus ^∞−∞log||𝑔(𝑡)||1+𝑡²d𝑡=−∞,(3.32) and, by Lemma 2.5, 𝑔(𝑧)≡0.

Acknowledgments

The author gratefully acknowledges the help of Prof. E. Zikkos to improve the original version of the paper. The author also thanks the referees for improvement of the paper. This paper supported by Natural Science Foundation of Yunnan Province in China (Grant no. 2009ZC013X) and Basic Research Foundation of Education Bureau of Yunnan Province in China (Grant no. 09Y0079).