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Abstract and Applied Analysis
Volume 2011, Article ID 605259, 8 pages
doi:10.1155/2011/605259
Research Article
Sharp Generalized Seiffert Mean Bounds for
Toader Mean
Yu-Ming Chu,1 Miao-Kun Wang,1 Song-Liang Qiu,2
and Ye-Fang Qiu1
1
2
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China
Correspondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn
Received 4 June 2011; Revised 10 August 2011; Accepted 11 August 2011
Academic Editor: Detlev Buchholz
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.
For p ∈ 0, 1, the generalized Seiffert mean of two positive numbers a and b is defined by Sp a, b pa − b/ arctan2pa − b/a b, 0 < p ≤ 1, a /
b; a b/2, p 0, a /
b; a, a b. In this paper,
we find the greatest value α and least value β such that the double inequality Sα a, b < T a, b <
Sβ a, b holds for all a, b > 0 with a /
b, and give new bounds for the complete elliptic integrals of
π/2 a2 cos2 θ b2 sin2 θdθ denotes the Toader mean of two
the second kind. Here, T a, b 2/π 0
positive numbers a and b.
1. Introduction
For p ∈ 0, 1, the generalized Seiffert mean of two positive numbers a and b is defined by
⎧
pa − b
⎪
⎪
,
⎪
⎪
⎪
arctan
2pa
− b/a b
⎪
⎪
⎪
⎪
⎨
Sp a, b a b
⎪
,
⎪
⎪
2
⎪
⎪
⎪
⎪
⎪
⎪
⎩a,
0 < p ≤ 1, a / b,
p 0, a /
b,
1.1
a b.
It is well known that Sp a, b is continuous and strictly increasing with respect to p ∈
0, 1 for fixed a, b > 0 with a /
b. In particular, if p 1/2, then the generalized Seiffert mean
2
Abstract and Applied Analysis
reduces to the Seiffert mean
Sa, b ⎧
⎪
⎨
a−b
,
2 arctana − b/a b
⎪
⎩a,
a/
b,
a b.
1.2
Recently, the Seiffert mean and its generalization have been the subject of intensive
research, many remarkable inequalities for these means can be found in the literature 1–5.
In 6, Toader introduced the Toader mean T a, b of two positive numbers a and b as
follows:
2 π/2
a2 cos2 θ b2 sin2 θdθ,
π 0
⎧
2
⎪
⎪
1 − b/a
⎪
⎪ 2aE
⎪
⎪
⎪
, a > b,
⎪
⎪
⎨
π
2bE
1 − a/b2
⎪
⎪
⎪
⎪
, a < b,
⎪
⎪
⎪
π
⎪
⎪
⎩
a,
a b,
T a, b 1.3
π/2
1/2
where Er 0 1 − r 2 sin2 t dt, r ∈ 0, 1 is the complete elliptic integral of the second
kind.
Vuorinen 7 conjectured that
1.4
M3/2 a, b < T a, b
for all a, b > 0 with a /
b, where
⎧
⎪
ap bp 1/p
⎪
⎨
,
2
Mp a, b ⎪
⎪
⎩√ab,
p/
0,
1.5
p0
is the power mean of order p of two positive numbers a and b. This conjecture was proved
by Barnard et al. 8.
In 9, Alzer and Qiu presented a best possible upper power mean bound for the
Toader mean as follows:
T a, b < Mlog 2/ logπ/2 a, b
1.6
for all a, b > 0 with a /
b.
The main purpose of this paper is to find the greatest value α and least value β such
b and
that the double inequality Sα a, b < T a, b < Sβ a, b holds for all a, b > 0 with a /
give new bounds for the complete elliptic integrals of the second kind.
Abstract and Applied Analysis
3
2. Lemmas
In order to establish our main result, we need several formulas and lemmas, which we present
in this section.
The following formulas were presented in 10, Appendix E, pages 474-475: Let r ∈
0, 1, then
Kr π/2 1 − r 2 sin2 t
−1/2
dt,
0
Er π/2 1 − r 2 sin2 t
1/2
dt,
K0 π
,
2
K 1− ∞,
E0 π/2, E 1− 1,
0
dEr Er − Kr
dKr Er − 1 − r 2 Kr
,
,
2
dr
dr
r
r1 − r d Er − 1 − r 2 Kr
dKr − Er rEr
rKr,
,
dr
dr
1 − r2
√ 2Er − 1 − r 2 Kr
2 r
E
.
1r
1r
2.1
Lemma 2.1 see 10, Theorem 1.25. For −∞ < a < b < ∞, let fx, gx : a, b → R be
continuous on a, b and be differentiable on a, b, let g x / 0 for all x ∈ a, b. If f x/g x is
increasing (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.
Lemma 2.2. 1 Er − 1 − r 2 Kr/r 2 is strictly increasing from 0, 1 onto π/4, 1;
2 {Er − 1 − r 2 Kr/r 2 − π/4}/r 2 is strictly increasing from 0, 1 onto π/32,
1 − π/4;
3 Kr − Er/r 2 is strictly increasing from 0, 1 onto π/4, ∞;
4 2Er − 1 − r 2 Kr is strictly increasing from 0, 1 onto π/2, 2;
5 Fr 2 − r 2 Kr − 2Er/r 4 is strictly increasing from 0, 1 onto π/16, ∞;
6 Gr 4π − πr 2 − 8Er/r 4 is strictly increasing from 0, 1 onto 3π/16, 3π − 8.
Proof. Parts 1–4 can be found in 10, Theorem 3.211, Theorem 3.316, and Exercise
3.4311 and 13.
For part 5, clearly F1− ∞. Let F1 r 2 − r 2 Kr − 2Er and F2 r r 4 , then
Fr F1 r/F2 r, F1 0 F2 0 0 and
F1 r Er − 1 − r 2 K
.
F2 r
4r 2 1 − r 2 2.3
It follows from 2.3 and part 1 together with Lemma 2.1 that Fr is strictly
increasing in 0, 1 and F0 π/16.
4
Abstract and Applied Analysis
For part 6, clearly G1− 3π − 8. Let G1 r 4π − πr 2 − 8Er and G2 r r 4 , then
Gr G1 r/G2 r, G1 0 G2 0 0, and
G1 r
2 − r 2 Kr − 2Er
Er − 1 − r 2 Kr /r 2 − π/4
.
2G2 r
r2
r4
2.4
From 2.4, parts 2 and 5 together with Lemma 2.1, we know that Gr is strictly
increasing in 0, 1, and f0 3π/16.
√
√
Lemma 2.3. 1 gr arctan
3r/2
−
3πr/{42Er − 1 − r 2 Kr} is strictly increasing
√
√
from 0, 1 onto 0, arctan 3/2 − 3π/8.
2 fr arctan r − πr/{22Er − 1 − r 2 Kr} < 0 for r ∈ 0, 1.
√
√
Proof. For part 1, clearly g0 0 and g1− arctan 3/2 − 3π/8 0.0335 · · · > 0.
Simple computation leads to
√
√
3πEr
2 3
g r −
4 3r 2 42Er − 1 − r 2 Kr2
√
2.5
3r 4 Er
44 3r 2 2Er − 1 − r 2 Kr2
g1 r,
2
where g1 r {82Er − 1 − r 2 Kr − π4 3r 2 Er}/r 4 Er.
Making use of Lemma 2.2 1, 2, and 6, we get
2
16 Er − 1 − r 2 Kr /r 2 − π/4
Er − 1 − r 2 Kr
8
·
g1 r Er
r2
r2
4π − πr 2 − 8Er
r4
3π
16 π 2
π
·
− 3π − 8 8 −
> 0.
>
16 ·
π
4
32
2
2.6
−
Therefore, part 1 follows from 2.5 and 2.6 together with the limiting values of
gr at r 0 and r 1.
For part 2, simple computations yield that
lim fr lim− fr 0,
r → 0
f r 2.7
r →1
f1 r
21 r 2 2Er − 1 − r 2 Kr2
,
2.8
Abstract and Applied Analysis
5
2
where f1 r 22Er − 1 − r 2 Kr − π1 r 2 Er. Note that
lim f1 r 0,
2.9
lim f1 r 8 − 2π > 0,
2.10
r → 0
r → 1−
4 2Er − 1 − r 2 Kr Er − 1 − r 2 Kr
− 2πrEr
r
Er − Kr
−π 1 r 2
r
rf2 r,
f1 r 2.11
where f2 r 42Er−1−r 2 KrEr−1−r 2 Kr/r 2 −2πErπ1r 2 Kr−Er/r 2 .
From Lemma 2.21, 3, and 4 together with the monotonicity of Er we know that
f2 r is strictly increasing in 0, 1. Moreover,
π2
,
4
2.12
lim f2 r ∞.
2.13
lim f2 r −
r →0
r → 1−
Equations 2.11–2.13 and the monotonicity of f2 r lead to the conclusion that there
exists r0 ∈ 0, 1 such that f1 r is strictly decreasing in 0, r0 and strictly increasing in r0 , 1.
It follows from 2.8–2.10 and the piecewise monotonicity of f1 r that there exists
r1 ∈ 0, 1 such that fr is strictly decreasing in 0, r1 and strictly increasing in r1 , 1.
Therefore, part 2 follows from 2.7 and the piecewise monotonicity of fr.
3. Main Result
Theorem 3.1. Inequality S√3/4 a, b < T a, b < S1/2 a, b holds for all a, b > 0 with a /
b, and
S√3/4 a, b and S1/2 a, b are the best possible lower and upper generalized Seiffert mean bounds for
the Toader mean T a, b, respectively.
Proof. Firstly, we prove that
S√3/4 a, b < T a, b < S1/2 a, b
for all a, b > 0 with a /
b.
3.1
6
Abstract and Applied Analysis
Without loss of generality, we assume that a > b. Let t b/a < 1, r 1 − t/1 t.
Then 1.1 and 1.3 lead to
T a, b − S√3/4 a, b √
3a1 − t
2a E
1 − t2 −
√
π
4 arctan 31 − t/21 t
√
√ 2a
3ar
2 r
E
−
√
π
1r
21 r arctan
3/2 r
√
3ar
2a 2Er − 1 − r 2 Kr
−
√
π
1r
21 r arctan
3/2 r
3.2
2a 2Er − 1 − r 2 Kr
√
gr,
π1 r arctan
3/2 r
a1 − t
2a E
1 − t2 −
π
2 arctan1 − t/1 t
√ 2 r
ar
2a
E
−
π
1r
1 r arctan r
T a, b − S1/2 a, b ar
2a 2Er − 1 − r 2 Kr
−
π
1r
1 r arctan r
3.3
2a 2Er − 1 − r 2 Kr
fr,
π1 r arctan r
where gr and fr are defined as in Lemma 2.3.
Therefore, inequality 3.1 follows from 3.2 and 3.3 together with Lemma 2.3.
Next, we prove that S√3/4 a, b and S1/2 a, b are the best possible lower and upper
generalized Seiffert mean bounds for the Toader mean T a, b, respectively.
For any ε > 0 and 0 < x < 1, from 1.1 and 1.3 one has
lim S1/2−ε 1, x − T 1, x x→0
2
1
2
1 − 2ε
− <
− 0,
2 arctan1 − 2ε π 2 arctan 1 π
S√3/4ε 1, 1 − x − T 1, 1 − x Jx
,
√
arctan
3 4ε x/22 − x
√
√
√
where Jx 3/4 εx − 2E 2x − x2 arctan{ 3 4εx/22 − x}/π.
3.4
3.5
Abstract and Applied Analysis
7
Letting x → 0 and making use of Taylor expansion, we get
√
√
1
1
3
3
Jx ε x−
ε x 1 − x x2 o x2
4
4
2
16
⎧
⎫
⎡
√
2 ⎤
⎨
⎬
1
1
1
3
× 1 x⎣ −
ε ⎦x2 o x2
⎩
⎭
2
4 3
4
ε
3
√
3
ε
2
3.6
√
3
ε x3 o x3 .
4
Inequality 3.4 and equations 3.5 and 3.6 imply that for any ε > 0 there exist
δ1 δ1 ε > 0 and δ2 δ2 ε > 0, such that S√3/4ε 1, 1 − x > T 1, 1 − x for x ∈ 0, δ1 and
S1/2−ε 1, x < T 1, x for x ∈ 0, δ2 .
From Theorem 3.1, we get new bounds for the complete elliptic integrals of the second
kind as follows.
Corollary 3.2. The inequality
√
3π 1 − 1 − r 2
$
#√ √
√
8 arctan 3 1 − 1 − r 2 / 2 1 1 − r 2
√
√
π 1 − 1 − r2
< Er <
√
√
4 arctan 1 − 1 − r 2 / 1 1 − r 2
3.7
holds for all r ∈ 0, 1.
Acknowledgments
This research was supported by the Natural Science Foundation of China under Grant
11071069 and Innovation Team Foundation of the Department of Education of Zhejiang
Province under Grant T200924. The authors wish to thank the anonymous referees for their
careful reading of the manuscript and their fruitful comments and suggestions.
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Abstract and Applied Analysis
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