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Abstract and Applied Analysis
Volume 2011, Article ID 671765, 8 pages
doi:10.1155/2011/671765
Research Article
Monotonicity, Convexity, and Inequalities
Involving the Agard Distortion Function
Yu-Ming Chu,1 Miao-Kun Wang,1
Yue-Ping Jiang,2 and Song-Liang Qiu3
1
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
3
Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China
2
Correspondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn
Received 22 June 2011; Accepted 10 November 2011
Academic Editor: Martin D. Schechter
Copyright q 2011 Yu-Ming Chu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We present some monotonicity, convexity, and inequalities for the Agard distortion function ηK t
and improve some well-known results.
1. Introduction
For r ∈ 0, 1, Lengedre’s complete elliptic integrals of the first and second kind 1 are
defined by
K Kr π/2 1 − r 2 sin2 θ
−1/2
dθ,
0
K r K r ,
π
K1 ∞,
K0 ,
2
π/2 1/2
E Er 1 − r 2 sin2 θ
dθ,
0
E r E r ,
π
E0 ,
2
respectively. Here and in what follows, we set r √
E1 1,
1 − r2.
1.1
1.2
2
Abstract and Applied Analysis
Let μr be the modulus of the plan Grötzsch ring B2 \ 0, r for r ∈ 0, 1, where B2 is
the unit disk. Then, it follows from 2 that
μr π K r
.
2 Kr
1.3
For K ∈ 0, ∞, the Hersch-Pfluger distortion function ϕK r is defined as
ϕK r μ−1
μr
K
for r ∈ 0, 1,
ϕK 0 ϕK 1 − 1 0,
1.4
while the Agard distortion function ηK t and the linear distortion function λK are defined
by
ϕK r
,
ηK t ϕ1/K r λK ηK 1,
r
t
1t
t ∈ 0, ∞,
1.5
respectively.
It is well known that the functions ηK t and λK play a very important role in
quasiconformal theory, quasiregular theory, and some other related fields 3–8. For example,
Martin 8 found that the sharp upper bound in Schottky’s theorem can be expressed by
ηK t, and in 9–15 the authors established a number of remarkable properties for the Agard
distortion function ηK t.
In 14, the authors proved that
eπK−1 < λK < eaK−1 ,
1.6
ebK−1/K < λK < eπK−1/K
1.7
√
for all K ∈ 1, ∞, where a 4/πK1/ 22 4.3768 . . ., b a/2. Recently, Anderson et al.
15 established that
λK < eπb/KK−1 ,
1.8
elog 2a−log 2/KK−1 < λK < eπa−log 2/KK−1
1.9
for all K ∈ 1, ∞, where a and b are defined as in inequalities 1.6 and 1.7, respectively.
The purpose of this paper is to present the new monotonicity, convexity, and
inequalities for the Agard distortion function ηK t and improve inequalities 1.6–1.9.
Our main results are Theorems 1.1 and 1.2 as follows.
√
Theorem 1.1. Let K ∈ 1, ∞, a 4/πK1/ 22 4.3768 . . ., b a/2, and c ∈ R. Then, the
following statements are true.
1 fK λK/K c is strictly increasing from 1, ∞ onto 1, ∞ for c ≤ a; if c > a, then
there exists K0 ∈ 1, ∞, such that f is strictly decreasing in 1, K0 and strictly increasing
Abstract and Applied Analysis
3
in K0 , ∞. In particular, the inequality λK ≥ K c holds for all K ∈ 1, ∞ with the best
possible constant c a.
2 gK log ηK t − log t/K − 1 is convex in 1, ∞ for fixed t ∈ 0, ∞.
3 If t ≥ 1 and r t/1 t, then hK log ηK t − log t/K − 1/K is strictly
increasing from 1, ∞ onto 2KrK r/π, πKr/K r.
√
Theorem 1.2. Let t ∈ 0, ∞, r t/1 t, a 4/πK1/ 22 , b a/2, Ar π 2 /2μr,
Br 8KrK r2 Er − r 2 Kr/π 2 , and Fc K Klog ηK t − log t/K − 1 − c. Then,
the following statements are true.
1 Fc K is strictly decreasing from 1, ∞ onto −∞, 4KrK r/π − c for c > Ar.
If c Ar, then Fc K is strictly decreasing from 1, ∞ onto Ar − 4 log 2 −
log t, 4KrK r/π − Ar. Moreover,
teK−1ArAr−4 log 2−log t/K < ηK t < teK−1Ar4KrK r/π−Ar/K
1.10
for all t ∈ 0, ∞ and K ∈ 1, ∞. In particular, if t 1, then 1.10 becomes
eK−1ππ−4 log 2/K < λK < eK−1πa−π/K .
1.11
2 If c ≤ Br, then Fc K is strictly increasing from 1, ∞ onto 4KrK r/π − c, ∞.
Moreover,
ηK t > teK−1Br4KrK r/π−Br/K
1.12
for all t ∈ 0, ∞ and K ∈ 1, ∞. In particular, if t 1, then 1.12 becomes
λK > eK−1bb/K ebK−1/K .
1.13
3 If Br < c < Ar, then there exists K1 ∈ 1, ∞ such that Fc K is strictly decreasing on
1, K1 and strictly increasing on K1 , ∞.
4 Fc K is convex in 1, ∞.
2. Lemmas
In order to prove our main results, we need several formulas and lemmas, which we present
in this section.
4
Abstract and Applied Analysis
The following formulas
were presented in 14, Appendix E, pp. 474-475. Let t ∈
0, ∞, K ∈ 0, ∞, r t/1 t ∈ 0, 1, and s ϕK r. Then,
dKr Er − r 2 Kr
,
dr
rr 2
dEr Er − Kr
,
dr
r
π
KrE r K rEr − KrK r ,
2
dμr
π2
−
,
dr
4rr 2 Kr2
∂s ss 2 KsK s
,
∂r rr 2 KrK r
2
∂s
2
ss KsK s,
∂K πK
2
ϕK r2 ϕ1/K r 1,
ηK t 2
s
,
s
2.1
∂ηK t
4
2
ηK tKsK s K s2 ηK t.
∂K
πK
μr
Lemma 2.1 see 14, Theorem 1.25. For −∞ < a < b < ∞, let f, g : a, b → R be continuous
0 on a, b. If f x/g x is increasing
on a, b and differentiable on a, b, and let g x /
(decreasing) on a, b, then so are
fx − fa
,
gx − ga
fx − fb
.
gx − gb
2.2
If f x/g x is strictly monotone, then the monotonicity in the conclusion is also strict.
The following lemma can be found in 14, Theorem 3.211 and 7, Lemma 3.321
and Theorem 5.132.
Lemma 2.2. 1 Er − r 2 Kr/r 2 is strictly increasing from 0, 1 onto π/4, 1;
1/2;
2 r c Kr is strictly decreasing from 0, 1
√ onto 0, π/2 if and only if c ≥ √
3 KrK r is strictly decreasing in 0, 2/2 and strictly increasing in 2/2, 1;
4 μr log r is strictly decreasing from 0, 1 onto 0, log 4.
√
Lemma 2.3. Let r ∈ 1/ 2, 1, K ∈ 1, ∞, and s ϕK r. Then, GK ≡ {π/2Ks}2 μr/K s2 is strictly decreasing from 1, ∞ onto K r2 /Kr2 , π 2 /2Kr2 .
Proof. Clearly G1 π 2 /2Kr2 , G∞ K r2 /Kr2 . Differentiating GK, one has
G K 4
μr2 KsK s−2 E s − s2 K s
πK
π
2
− Ks−2 K s Es − s Ks
K
4
Ks−2 K s−2 G1 K,
πK
where G1 K E s − s2 K sKs3 μr2 − π 2 Es − s 2 KsK s3 /4.
2.3
Abstract and Applied Analysis
5
From Lemma 2.21 and 2, we clearly see that G1 K is strictly decreasing in 1, ∞.
Moreover,
π2 2
Er − r Kr K r3
lim G1 K E r − r 2 K r Kr3 μr2 −
K→1
4
π2 2
K r G2 r,
4
2.4
2 2
where G2 r KrE
√ r−r K r−K√rEr−r Kr is also strictly decreasing in 0, 1.
Thus, G2 r ≤ G2 2/2 0 for r ∈ 1/ 2, 1, and G1 K < G1 1 ≤ 0 for K ∈ 1, ∞.
Therefore, the monotonicity of GK follows from 2.3 and 2.4 together with the fact
that G1 K < 0 for K ∈ 1, ∞.
3. Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1. For√part 1, clearly f1 1. Let r μ−1 π/2K for K ∈ 1, ∞, then
λK r/r 2 , r ∈ 1/ 2, 1,
2 2
dr
rr K r2 ,
dK π
4
dλK
λKK r2 ,
dK
π
lim fK lim
K → ∞
r 2 K r
r → 1 r 2 Kr
3.1
∞.
3.2
Making use of 3.1, we have
K c1 f K
4
f1 K ≡ K rKr − c.
λK
π
3.3
It follows from Lemma 2.23 that f1 K is strictly increasing from 1, ∞ onto a −
c, ∞. Then, from 3.2 and 3.3, we know that f is strictly increasing from 1, ∞ onto 1, ∞
for c ≤ a. If c > a, then there exists K0 ∈ 1, ∞ such that f is strictly decreasing in 1, K0 and
strictly increasing in K0 , ∞. Moreover, the inequality λK ≥ K c holds for all K ∈ 1, ∞
with the best possible constant
c a.
For part 2, denote r t/1 t. Differentiating gK, we get
g K 2K s2 K − 1/μr − log ηK t − log t
K − 12
.
3.4
Let g1 K 2K s2 K − 1/μr − log ηK t − log t and g2 K K − 12 , then
g1 1 g2 1 0, g K g1 K/g2 K and
g1 K
2 2 K s3 .
E
K
g
≡
−
−
s
K
s
s
3
g2 K
μr2
3.5
6
Abstract and Applied Analysis
Clearly, g3 K is strictly increasing in 1, ∞. Then, 3.5 and Lemma 2.1 lead to the conclusion
that g K is strictly increasing in 1,√∞. Therefore, gK is convex in 1, ∞.
For part 3, if t ≥ 1, then r ≥ 2/2. Let h1 K log ηK t − log t and h2 K K − 1/K,
then h1 1 h2 1 0, hK h1 K/h2 K, and
2
h1 K 2K s /μr 2μr
,
GK
1 K −2
h2 K
3.6
where GK is defined as in Lemma 2.2.
Therefore, hK is strictly increasing in 1, ∞ for t ≥ 1 follows from Lemmas 2.1
and 2.2 together with 3.6. Moreover, making use of l’Hôpital’s rule, we have h1 2KrK r/π, h∞ πKr/K r.
Proof of Theorem 1.2. Differentiating Fc K gives
⎡
⎤
2
/μr
−
log
η
2K
−
1
−
log
t
s
K
t
K
log ηK t − log t
⎢
⎥
− c K⎣
Fc K ⎦
2
K−1
K − 1
2K s KK − 1/μr − log ηK t − log t
2
K − 12
3.7
− c.
Let
HK 2K s2 KK − 1/μr − log ηK t − log t
K − 12
,
3.8
H1 K 2K s2 KK − 1/μr − log ηK t − log t, and H2 K K − 12 , then HK H1 K/H2 K, H1 1 H2 1 0, and
H1 K
4 2
K s3 .
Es
−
s
H
Ks
≡
K
3
πμr
H2 K
3.9
Clearly, that H3 K is strictly increasing in 1, ∞ follows from Lemma 2.21 and 2.
Then, from 3.8 and 3.9 together with Lemma 2.1, we know that HK is strictly increasing
in 1, ∞. Moreover, l’Hôpital’s rule leads to
lim HK Br,
K→1
lim HK Ar.
K→∞
3.10
For part 1, if c > Ar, then from 3.7 and 3.8, we know that Fc K < 0 for K ∈
1, ∞ and Fc K is strictly decreasing in 1, ∞. Moreover,
lim Fc K 4KrK r/π − c,
K→1
lim Fc K −∞.
K→∞
3.11
Abstract and Applied Analysis
7
If c Ar, then Fc K is also strictly decreasing in 1, ∞ and Fc 1 4KrK r/π −
Ar, and from Lemma 2.24 we get
K −2 log s − 2μ s Ar − log t
K→∞K − 1
lim Fc K lim
K→∞
3.12
Ar − 4 log 2 − log t.
Therefore, inequalities 1.10 and 1.11 follows from 3.12 and the monotonicity of
Fc K when c Ar.
For part 2, if c ≤ Br, then that Fc K is strictly increasing in 1, ∞ follows from
3.7 and 3.8. Note that
lim Fc K 4KrK r/π − c,
K→1
lim Fc K ∞.
K→∞
3.13
Therefore, inequalities 1.12 and 1.13 follow from 3.13 and the monotonicity of
Fc K when c Br.
For part 3, if Br < c < Ar, then from 3.7 and 3.8 together with the
monotonicity of HK we clearly see that there exists K1 ∈ 1, ∞, such that Fc K < 0 for
K ∈ 1, K1 and Fc K > 0 for K ∈ K1 , ∞. Hence, Fc K is strictly decreasing in 1, K1 and
strictly increasing in K1 , ∞.
Part 4 follows from 3.7 and 3.8 together with the monotonicity of HK.
Taking t 1 in Theorem 1.2, we get the following corollary.
Corollary 3.1. Let a and b be defined as in Theorem 1.2, c ∈ R, and fc K K{log λK/K −
1 − c}. Then,
1 if c > π, then fc K is strictly decreasing from 1, ∞ onto −∞, a − c; if c π, then
fc K is strictly decreasing from 1, ∞ onto π − 4 log 2, a − π;
2 if c ≤ b, then fc K is strictly increasing from 1, ∞ onto a − c, ∞;
3 if b < c < π, then there exists K2 ∈ 1, ∞, such that fc K is strictly decreasing in 1, K2 and strictly increasing in K2 , ∞;
4 fc K is convex in 1, ∞.
Inequalities 1.11 and 1.13 lead to the following corollary, which improve inequalities 1.6–1.9.
Corollary 3.2. Let a and b be defined as in Theorem 1.2, then the following inequality
max eK−1ππ−4 log 2/K , ebK−1/K < λK < eK−1πa−π/K .
holds for all K ∈ 1, ∞.
3.14
8
Abstract and Applied Analysis
Acknowledgments
This research was supported by the Natural Science Foundation of China Grants nos.
11071059, 11071069, and 11171307 and the Innovation Team Foundation of the Department
of Education of Zhejiang Province Grant no. T200924.
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