Rn中Gr?tzsch环的共形模的一个不等式,该文是不等式毕业论文格式范文跟不等式和tzsch类论文参考文献范文.
不等式论文参考文献：
QIU Songliang, WU Haiqin
(School of Sciences, Zhejiang SciTech University, Hangzhou 310018, China)
Abstract： Let r′等于1－r2 and Mn(r) be the (conformal) modulus of the Grtzsch Ring in the quasiconformal theory in Rn, for n≥3 and r∈(0,1). In this paper, a double inequality is obtained for the function H(r)≡r′2Mn(r)Mn(r′)n－1+r2Mn(r′)Mn(r)n－1, thus improving known bounds for H(r), and correcting an error in the proof of a related inequality for H(r) which was given in a monograph by G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen.
Key words： ndimensional quasiconformal theory; the Grtzsch ring; modulus; inequalities
CLC number： O174.55
Document code： A
文章编号： 1673\|3851 (2018) 01\|0103\|04
0Notation and Main Results
For n≥2, let Rn denote the ndimensional Euclidian space, n等于Rn∪{∞},Bn the unit ball in Rn, and let e1, e2,..., en be the standard unit vectors in Rn. A domain Dn is said to be a ring domain(or a ring in brief) if n＼D consists of two components C0 and C1, where C0 is bounded. Such a ring is usually denoted by R(C0,C1). For s>1, the socalled Grtzsch ring is defined by
RG,n(s)等于R(Bn,［se1,∞］),s>1,
which means that the complementary components of the Grtzsch ring RG,n(s) with respect to n are C0等于Bn等于Bn∪Bn and C1等于［se1,∞］. (See ［1, p.149］.)
For E,FGn, we denote the family of curves joining E and F in G by Δ(E,F;G). If G等于Rn or n, then we may omit G and simply denote Δ(E,F;G) by Δ(E,F). Let Γ be a family of curves in n, 等于R∪{∞}, and for an arbitrary locally rectifiable curve γ∈Γ, put F(Γ)等于{ρ|ρ:Rn→ is a nonnegative Borelmeasurable function such that ∫γρds≥1}. The function ρ is said to be admissible if ρ∈F(Γ). The modulus of Γ is then defined as
M(Γ)等于infρ∈F(Γ)∫Rnρndm,
where m is the ndimensional Lebesgue measure. By ［1, Theorem 8.28, (8.31), (8.34) and (8.35)］, the conformal capacity cap RG,n(s) of the Grtzsch ring RG,n(s) can be expressed by
γn(s)≡capRG,n(s)≡M(Δ(Bn,［se1,∞］)),
while the (conformal) modulus of RG,n(1/r) is defined by
Mn(r)等于modRG,n(1/r)等于ωn－1γn(1/r)1/(n－1),r∈(0,1),
where ωn－1 is the surface area of the unit sphere Sn－1等于Bn. Clearly, μ(r)≡M2(r) is exactly the socalled Grtzsch ring function, which has the following expression
μ(r)等于π2K′(r)K(r),(1)
where
K(r)等于∫π/20dt1－r2sin2t and K′(r)等于K(r′)
for r∈(0,1), are the complete elliptic integrals of the first kind (see ［1］ or ［2］). Here and hereafter, we always let r′等于1－r2 for r∈［0,1］. It is well known that the Grtzsch ring RG,n(1/r) and its modulus Mn(r) or its capacity γn(1/r) play an extremely important role in the study of quasiconformal mappings in Rn.
The Grtzsch ring constant λn is defined by
logλn等于limr→0+［Mn(r)+logr］,
which is indispensable in the study of Mn(r) and γn(s). It is well known that λ2等于4. Unfortunately, so far we he only known some estimates for λn when n≥3, among which is the following double inequality
2e0.76(n－1)<λn≤2en+(1/n)－(3/2),n≥3(2)
(see ［1, Theorem 12.21(1)］ and ［3］).
Now we introduce the gamma and beta functions, and some constants depending only on n, which are needed in the study of the properties of Mn(r) and γn(s). As usual, for complex numbers x and y with Re x>0 and Re y>0, the gamma and beta functions are defined by
Γ(x)等于∫∞0tx－1e－tdt and B(x,y)等于∫10tx－1(1－t)y－1dt,
respectively. (Cf. ［4］ and ［5］.) It is well known that, for n≥3, the volume Ωn of Bn and the (n－1)dimensional surface area ωn－1 of Sn－1can be expressed by
Ωn等于2πnΩn－2等于πn/2Γ(1+n/2) and ωn－1等于nΩn等于nπn/2Γ(1+n/2),
respectively. (Cf. ［1, 2.23］ and ［6］.) Let
Jn等于∫π/20(sint)(2－n)/(n－1)dt等于12B12(n－1),12,
cn等于(2Jn)1－nωn－2,An等于ωn－12ncn1/(n－1).
In particular,
J2等于π/2,J3等于2K(1/2)等于2.62205…,
c2等于2/π,c3等于4π2Γ(1/4)－4等于0.22847…,
A2等于π2/4 and A3等于J3.
Some properties of Ωn, ωn－1, Jn, cn and An were given in ［1, pp.3844&163］ and in ［6］.
In the sequel, we let arth denote the inverse function of the hyperbolic tangent tanh, that is,
arthx等于12log1+x1－x, －1<x<1.
During the past decades, many properties he been obtained for μ(r) (cf. ［1］－［2］ and ［7］). The known properties of Mn(r), however, are much less than those of μ(r), because of lack of effective tools for the study of Mn(r) when n≥3. For example, we he no explicit expression as or similar to (1) for Mn(r) when n≥3. For the known properties of Mn(r) and its related functions, the reader is referred to ［1］, ［3］ and ［713］. Some of these known results for Mn(r) are related to the constants λn,Ωn, ωn－1, Jn, cn and An. For example, the following inequalities hold
An12μ1－r1+r1/(1－n)≤Mn(r)≤An12log1－r1+r1/(1－n)(3)
log1+r′r<Mn(r)<logλn(1+r′)2r(4)
0<Mn(r)n-1log1 + r1-r<2An-1n(5)
for r∈(0,1) and n≥3 (see ［1, Theorems 11.20(1), 11.21(2)&(4), and 11.21(5) ］).
On the other hand, if we let hn(r)等于r′2Mn(r)Mn(r′)n－1, then for all r∈(0,1),
h2(r)+h2(r′)等于μ(r)μ(r′)≡π2/4
by ［1, (5.2)］. It is well known that for each n≥2, all r∈(0,1) and for all K>0,
φK,n(r)2+φ1/K,n(r′)2等于1Mn(r)Mn(r′)等于const,
where φK,n(r)等于M-1n(αMn(r)) and α等于K1/(1－n) (cf. ［1, 8.70］). Therefore, it is quite significant for us to study the properties of the function hn, in order to reveal the properties of Mn(r) and φK,n(r). In ［8, Theorem 5.1(3)］, it was proved that for each n≥2 and all r∈(0,1),
An-1n等于ωn-12ncn<hn(r)+hn(r′)<4ωn-12ncnlogλn等于4An-1nlogλn(6)
Later, ［1, 11.36(2)］ says that for each n≥2 and all r∈(0,1),
An-1n等于ωn-1(2ncn)<hn(r)+hn(r′)<2An-1nlogλn(7)
However, the proof of the second inequality in (7) given in ［1, p.244］ contains an error. This proof in ［1, p.244］ is as follows: ［1, Corollary 11.23(1) and (4) ］ yield
hn(r)≤An-1nr′2log(λn/r)log(1/r),
and the upper bound in (7) follows, since ［1, Theorem 1.25 ］ implies that the function
r→r′2log(λn/r)log(1/r)
is increasing from (0,1) onto (1,2 logλn). It is easy to see that by this “proof ”, one can only obtain the following inequality
hn(r)<2An-1nlogλn,
so that the upper bound for hn(r)+hn(r′), which we can obtain by this method, is as follows
hn(r)+hn(r′)<4An-1nlogλn,
consisting with that in (6). So far, the known best upper bound for hn(r)+hn(r′) is given by (6).
In addition to indicating the error in the proof of (7) given in ［1, p.244］ as abovementioned, the main purpose of this paper is to improve the upper bound given in (6) by proving the following result.
Theorem 1Let hn(r)等于r′2Mn(r)Mn(r′)n－1. Then for each n≥2 and all r∈(0,1),
An-1n<hn(r) + hn(r′)<βAn-1nlogλn(8)
where
β等于1log(1+2)1+log(1+2)－log21.52+log2等于1.23108….
1Proof of Theorem 1
The proof of Theorem 1 stated in Section 0 requires the following lemma.
1.1A Technical Lemma
Lemma 1a) For r∈(0,1), let g(r)等于r2/arth r and f(r)等于g′(r)/r. Then f is strictly decreasing from (0,1) onto (－∞,∞).
b) The function F(r)≡g(r)+g(r′) is strictly increasing on 0,12, and decreasing on 12,1. In particular, for all r∈(0,1),
F(r)≤F12等于1log(1+2)(9)
The first equality in (9) holds if and only if r等于1/2.
Proof:a) Differentiation gives
g′(r)等于r(arthr)22arthr－rr′2,
so that
f(r)等于g′(r)r等于2arthr－r(r′arthr)2(10)
Clearly, f(0+)等于∞ and f(1－)等于－∞. By differentiation,
r′r(r′arthr)3f′(r)等于21－arthrr－r′2arthrr(11)
which is negative for all r∈(0,1) since the function r→(arthr)/r is strictly increasing from (0,1) onto (1,∞). This yields the result for f.
b) It is easy to verify that
1rF′(r)等于h(r)≡f(r)－f(r′).
By part (1), h is strictly decreasing from (0,1) onto (－∞,∞) and has a unique zero r0等于1/2 on (0,1). This yields the piecewise monotonicity of F.
Then the remaining conclusions are clear.
1.2Proof of Theorem 1
The first inequality in (8) was proved in ［8, Theorem 5.1(3)］.
Let H(r)等于hn(r)+hn(r′), and F be as in Lemma 1 b). By (5), we see that
Mn(r)n-1arthr<An-1n,n≥2,0<r<1(12)
On the other hand, the following inequality holds
Mn(r)<log(λn/2)+arthr′(13)
for each n≥2 and all 0<r<1, since the function
r→Mn(r)/［log(λn/2)+arthr′］
is strictly decreasing from (0,1) onto (0,1) by ［1, Theorem 11.21(4)］. It follows from (12) and (13) that
H(r)等于r′2Mn(r)arthr′·Mn(r′)n－1arthr′+r2Mn(r′)arthr·Mn(r)n－1arthr
≤An-1nr′2arthr′Mn(r)+r2arthrMn(r′)
≤An-1nr′2arthr′logλn2+arthr′+r2arthrlogλn2+arthr
等于An-1n1+r2arthr+r′2arthr′logλn2
等于An-1n1+F(r)logλn2.
This, together with Lemma 1 b), yields
H(r)≤An-1n1+1log(1+2)logλn2
等于An-1nlogλnlog(1+2)1+log(1+2)-log2logλn
(14)
By (2), the following double inequality holds
1logλn<10.76(n－1)+log2≤11.52+log2(15)
with equality if and only if n等于3. Since log(1+2)－log2等于0.188226…>0, it follows from (14) and (15) that
H(r)≤βAn-1nlogλn,
where
β等于1log(1+2)1+log(1+2)－log21.52+log2等于1.23108….
This yields the second inequality in (8) as desired.
References：
［1］ Anderson G D, Vamanamurthy M K, Vuorinen M. Conformal Invariants, Inequalities, and Quasiconformal Maps［M］. New York: John Wiley and Sons,1997.
［2］ Ahlfors L V. Lectures on Quasiconformal Mappings［M］. 2nd ed. American Mathematical Society,2005.
［3］ Anderson G D, Frame J S. Numerical estimates for a Grtzsch ring constant［J］. Constr Approx,1988,4:223242.
［4］ Abramowitz M, Stegun I A(Eds.). Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables［M］. New York: Dover,1965.
［5］ Qiu S L, Vuorinen M. Handbook of Complex Analysis: Special Function in Geometric Function Theory: Volume 2［M］. Elsevier B V,2005:621659.
［6］ Qiu S L, Vuorinen M. Some properties of the gamma and psi functions with applications［J］. Math Comput,2005,74(250):723742.
［7］ Qiu S L. Grtzsch ring and Ramanujans modular equations［J］. Acta Mathematica Sinica,2000,43(2):283290.
［8］ Anderson G D, Qiu S L, Vamanamurthy M K. Grtzsch ring and quasiconformal distortion functions［J］. Hokkaido Math J,1995,24(3):551566.
［9］ Anderson G D, Vamanamurthy M K, Vuorinen M. Conformal invariants, quasiconformal maps, and special functions［M］//Quasiconformal Space Mappings. BerlinHeidelberg: SpringerVerlag,1992:119.
［10］ Anderson G D, Vamanamurthy M K, Vuorinen M. Inequalities for quasiconformal mappings in space［J］. Pacific J Math,1993,160:118.
［11］ Ikoma K. An estimate for the modulus of the Grtzsch ring in nspace［J］. Bull Yamagata Univ Natur Sci,1967,6:395400.
［12］ Qiu S L, Vamanamurthy M K. Elliptic integrals and the modulus of Grtzsch ring［J］. PanAmer Math J,1995,5(2):4160.
［13］ Vuorinen M. On the boundary behior of locally Kquasiconformal mappings in space［J］. Ann Acad Sci Fenn Ser A I,1980,5:7995.
Rn中Grtzsch环的共形模的一个不等式
裘松良,武海琴
（浙江理工大学理学院,杭州 310018）
摘 要： 设Mn(r)为n维拟共形理论中的Grtzsch环RG,n(1/r)的模,r′等于1－r2,其中0<r<1, n≥3.建立了函数H(r)≡r′2Mn(r)Mn(r′)n－1+r2Mn(r′)Mn(r)n－1满足的一个双向不等式,较大程度地改进了H(r)的已知上界,指出并纠正了G. D. Anderson、M. K. Vamanamurthy和M. Vuorinen的专著中给出的关于H(r)的一个上界的证明中存在的错误.
关键词： n维拟共形理论；Grtzsch环；共形模；不等式
(责任编辑： 康锋)
上文结束语:此文为关于不等式和tzsch方面的不等式论文题目、论文提纲、不等式论文开题报告、文献综述、参考文献的相关大学硕士和本科毕业论文.
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