Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 917120, 7 pages
doi:10.1155/2012/917120
Research Article
Optimal Bounds for Seiffert Mean in terms of
One-Parameter Means
Hua-Nan Hu,1 Guo-Yan Tu,2 and Yu-Ming Chu3
1
Acquisitions & Cataloging Department of Library, Huzhou Teachers College, Huzhou 313000, China
Department of Basic Course Teaching, Tongji Zhejiang College, Jiaxing 314000, China
3
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2
Correspondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn
Received 11 July 2012; Accepted 7 September 2012
Academic Editor: Francisco J. Marcellán
Copyright q 2012 Hua-Nan Hu 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.
The authors present the greatest value r1 and the least value r2 such that the double inequality
Jr1 a, b < T a, b < Jr2 a, b holds for all a, b > 0 with a / b, where T a, b and Jp a, b denote the
Seiffert and pth one-parameter means of two positive numbers a and b, respectively.
1. Introduction
For p ∈ R the pth one-parameter mean Jp a, b and the Seiffert mean T a, b of two positive
real numbers a and b are defined by
⎧ ⎪
p ap1 − bp1
⎪
⎪
,
p
⎪
⎪
p
⎪
⎪
⎪ p 1 a − b ⎪
⎪
⎪
a−b
⎪
⎨
,
Jp a, b log a − log b
⎪
⎪
⎪
⎪
⎪
ab log a − log b
⎪
⎪
⎪
,
⎪
⎪
a−b
⎪
⎪
⎩a,
⎧
⎨
a/
b, p /
0, −1,
a/
b, p 0,
1.1
a/
b, p −1,
a b,
a−b
,
T a, b 2 arctana − b/a b
⎩
a,
a/
b,
a b,
1.2
2
Journal of Applied Mathematics
respectively. Recently, both mean values have been the subject of intensive research. In
particular, many remarkable inequalities and properties for Jp and T can be found in the
literature 1–14.
It is well known that the one-parameter mean Jp a, b is continuous and strictly
increasing with respect to p ∈ R for fixed a, b > 0 with a / b. Many mean values are the
special case of the one-parameter mean, for example:
a b
,
J1 a, b 2
√
a ab b
,
J1/2 a, b √3
J−1/2 a, b ab,
2ab
J−2 a, b ,
a b
the arithmetic mean,
the Heronian mean,
1.3
the geometric mean,
the harmonic mean.
Seiffert 4 proved that the double inequality
M1 a, b < T a, b < M2 a, b
1.4
√
holds for all a, b > 0 with a /
b, where Mr a, b ar br /21/r r /
0 and M0 a, b ab
is the rth power mean of a and b.
In 15–17, the authors presented the best possible bounds for the Seiffert mean in
terms of the Lehmer, power-type Heron, and one-parameter Gini means as follows:
L0 a, b < T a, b < L1/3 a, b,
Hlog 3/ logπ/2 a, b < T a, b < H5/2 a, b,
1.5
S1 a, b < T a, b < S5/3 a, b.
for all a, b > 0 with a Lr a, b ar1 br1 /ar br , Hk a, b ak abk/2 / b, where √
1/k
1/α−2
k
α b /3 k / 0 and H0 a, b ab, and Sα a, b aα−1 bα−1 /a b
/ 2 and
1/ab
denote the Lehmer, power-type Heron, and one-parameter Gini means
S2 a, b aa bb of a and b, respectively.
The purpose of this paper is to answer the question: what are the greatest value r1 and
the least value r2 such that the double inequality
Jr1 a, b < T a, b < Jr2 a, b
holds for all a, b > 0 with a /
b?
2. Lemma
In order to establish our main result we need the following lemma.
1.6
Journal of Applied Mathematics
3
Lemma 2.1. If p 2/π − 2 1.75 · · · , t ≥ 1 and gt −p − 1t2p2 p 1t2p pp 1tp3 −
2p 12 tp2 2pp 3tp1 − 2p 12 tp pp 1tp−1 p 1t2 − p − 1, then there exists
λ ∈ 1, ∞ such that gt > 0 for t ∈ 1, λ and gt < 0 for t ∈ λ, ∞.
Proof. Let g1 t g t/t, g2 t t4−p g1 t and g3 t t4−p g2 5 t/4p2 p − 12 p 12 . Then
simple computations lead to
g1 0,
2.1
lim gt −∞,
2.2
t → ∞
g1 t − 2 p − 1 p 1 t2p 2p p 1 t2p−2 p p 1 p 3
2 × tp1 − 2 p 1 p 2 tp 2p p 1 p 3 tp−1
2.3
2
− 2p p 1 tp−2 p p 1 p − 1 tp−3 2 p 1 ,
g1 1 0,
2.4
lim g1 t −∞,
2.5
t → ∞
2
g2 t − 4p p − 1 p 1 tp3 4p p 1 p − 1 tp1 p p 1
2 × p 3 t4 − 2p p 1 p 2 t3 2p p − 1 p 1 p 3
2.6
2 × t2 − 2p p 1 p − 2 t p p 1 p − 1 p − 3 ,
g2 1 0,
2.7
lim g2 t −∞,
2.8
t → ∞
2 g2 t − 4p p − 1 p 1 p 3 tp2 4p p 1 p − 1 tp
2 2 4p p 1 p 3 t3 − 6p p 1 p 2 t2
2.9
2 4p p − 1 p 1 p 3 t − 2p p 1 p − 2 ,
g2 1 0,
lim g t
t → ∞ 2
−∞,
2.10
2.11
2
g2 t − 4p p − 1 p 1 p 2 p 3 tp1 4p2 p 1
2 2
× p − 1 tp−1 12p p 1 p 3 t2 − 12p p 1
× p 2 t 4p p − 1 p 1 p 3 ,
2
g2 1 12p 2 − p p 1 > 0,
lim g t
t → ∞ 2
−∞,
2.12
2.13
2.14
4
Journal of Applied Mathematics
2 2
g2 t − 4p p − 1 p 1 p 2 p 3 tp 4p2 p 1
2
2 × p − 1 tp−2 24p p 1 p 3 t
2.15
2 − 12p p 1 p 2 ,
2
g2 1 12p 2 − p 2p 3 p 1 > 0,
lim g t
t → ∞ 2
−∞,
2 2
4
g2 t − 4p2 p − 1 p 1 p 2 p 3 tp−1 4p2 p 1
2 2 × p − 1 p − 2 tp−3 24p p 1 p 3 ,
2 4
g2 1 8p p 1 −4p3 2p2 5p 9
2 > 8p p 1 −4 × 1.83 2 × 1.752 5 × 1.75 9
2.16
2.17
2.18
2.19
2
4.376p p 1 > 0,
4
lim g t
t → ∞ 2
−∞,
g3 t − p 2 p 3 t2 p − 2 p − 3
≤− p2 p3 p−2 p−3
2.20
2.21
−10p < 0
for t ∈ 1, ∞.
4
From the inequality 2.21 we clearly see that g2 t is strictly decreasing in 1, ∞.
4
Then 2.19 and 2.20 lead to the conclusion that there exists λ1 > 1 such that g2 t > 0
4
for t ∈ 1, λ1 and g2 t < 0 for t ∈ λ1 , ∞. Hence, g2 t is strictly increasing in 1, λ1 and
strictly decreasing in λ1 , ∞.
It follows from 2.16 and 2.17 together with the monotonicity of g2 t that there
exists λ2 > 1 such that g2 t > 0 for t ∈ 1, λ2 and g2 t < 0 for t ∈ λ2 , ∞. Therefore, g2 t is
strictly increasing in 1, λ2 and strictly decreasing in λ2 , ∞.
From 2.13 and 2.14 together with the monotonicity of g2 t we know that there
exists λ3 > 1 such that g2 t > 0 for t ∈ 1, λ3 and g2 t < 0 for t ∈ λ3 , ∞. So, g2 t is strictly
increasing in 1, λ3 and strictly decreasing in λ3 , ∞.
Equations 2.10 and 2.11 together with the monotonicity of g2 t imply that there
exists λ4 > 1 such that g2 t > 0 for t ∈ 1, λ4 and g2 t < 0 for t ∈ λ4 , ∞. Hence, g2 t is
strictly increasing in 1, λ4 and strictly decreasing in λ4 , ∞.
It follows from 2.7 and 2.8 together with the monotonicity of g2 t that there exists
λ5 > 1 such that g2 t > 0 for t ∈ 1, λ5 and g2 t < 0 for t ∈ λ5 , ∞. Therefore, g1 t is strictly
increasing in 1, λ5 and strictly decreasing in λ5 , ∞.
From 2.4 and 2.5 together with the monotonicity of g1 t we clearly see that there
exists λ6 > 1 such that g1 t > 0 for t ∈ 1, λ6 and g1 t < 0 for t ∈ λ6 , ∞. So, gt is strictly
increasing in 1, λ6 and strictly decreasing in λ6 , ∞.
Journal of Applied Mathematics
gt.
5
Therefore, Lemma 2.1 follows from 2.1 and 2.2 together with the monotonicity of
3. Main Result
Theorem 3.1. The double inequality
J2/π−2 a, b < T a, b < J2 a, b
3.1
holds for all a, b > 0 with a /
b, and J2/π−2 a, b and J2 a, b are the best possible lower and upper
one-parameter mean bounds for the Seiffert mean T a, b, respectively.
Proof. Without loss of generality, we assume that a > b. Let t a/b > 1. Then from 1.1 and
1.2 we have
b t2 t 1
t − 1 3 t2 − 1
4 arctan
−
.
J2 a, b − T a, b 6t 1 arctant − 1/t 1
t 1 t2 t 1
3.2
Let
t − 1 3 t2 − 1
ft 4 arctan
−
.
t 1 t2 t 1
3.3
Then simple computations lead to
f1 0,
f t t − 14
t2 1t2 t 1
2
> 0,
3.4
for t > 1.
Therefore, T a, b < J2 a, b for all a, b > 0 with a /
b follows from 3.2–3.4.
Next, we prove that
T a, b > J2/π−2 a, b
3.5
for all a, b > 0 with a /
b.
Let p 2/π − 2 1.75 · · · . Then 1.1 and 1.2 lead to
T a, b − Jp a, b
bp tp1 − 1
2 p 1 tp − 1 arctant − 1/t 1
p 1 t − 1tp − 1
t−1
.
− 2 arctan
t1
p tp1 − 1
3.6
6
Journal of Applied Mathematics
Let
p 1 t − 1tp − 1
t−1
.
− 2 arctan
Gt t1
p tp1 − 1
3.7
Then simple computations lead to
lim Gt lim Gt 0,
3.8
gt
G t ,
2
p1
p t − 1 t2 1
3.9
t→1
t → ∞
where gt is defined as in Lemma 2.1.
From Lemma 2.1 and 3.9 we know that there exists λ > 1 such that Gt is strictly
increasing in 1, λ and strictly decreasing in λ, ∞. Then 3.8 leads to that
Gt > 0,
3.10
for t > 1.
Therefore, the inequality 3.5 follows from 3.6, 3.7, and 3.10.
Finally, we prove that J2/π−2 a, b and J2 a, b are the best possible lower and upper
one-parameter mean bounds for the Seiffert mean T a, b, respectively.
Let p 2/π − 2, 0 < ε < 2 and x > 0. Then from 1.1 and 1.2 one has
Jpε x, 1
pε
p
π
π
× >
× 1,
x → ∞ T x, 1
pε1 2
p1 2
3.11
lim
T 1 x, 1 − J2−ε 1 x, 1 hx
,
2−ε
23 − ε 1 x − 1 arctanx/2 x
3.12
where
hx 3 − εx 1 x2−ε − 1 − 22 − ε 1 x3−ε − 1 arctan
x
.
2x
3.13
Letting x → 0 and making use of Taylor expansion we get
2 − ε1 − ε 2 ε1 − ε2 − ε 3
x −
x o x3
hx 3 − εx 2 − εx 2
6
3 − ε2 − ε 2 1 − ε2 − ε3 − ε 3
− 22 − ε 3 − εx x x o x3
2
6
x x2 x3
1
×
−
o x3
ε2 − ε3 − εx4 o x4 .
2
4
12
12
3.14
Journal of Applied Mathematics
7
The inequality 3.11 implies that for any 0 < ε < 2, there exists X Xε > 1, such that
T x, 1 < J2/π−2ε x, 1 for x ∈ X, ∞.
Equations 3.12–3.14 imply that for any 0 < ε < 2, there exists δ δε > 0 such that
T 1 x, 1 > J2−ε 1 x, 1 for x ∈ 0, δ.
Acknowledgments
This paper was supported by the Natural Science Foundation of Huzhou Teachers College
Grant no. 2010065 and the Natural Science Foundation of the Department of Education of
Zhejiang Province Grant no. Y201016277.
References
1 H. Alzer, “On Stolarsky’s mean value family,” International Journal of Mathematical Education in Science
and Technology, vol. 20, no. 1, pp. 186–189, 1987.
2 H. Alzer, “Über eine einparametrige Familie von Mittelwerten,” Bayerische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte, pp. 1–9, 1987.
3 H. Alzer, “Über eine einparametrige Familie von Mittelwerten. II,” Bayerische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte, pp. 23–29, 1988.
4 H. -J. Seiffert, “Aufgabe β16,” Die Wurzel, vol. 29, pp. 221–222, 1995.
5 F. Qi, “The extended mean values: definition, properties, monotonicities, comparison, convexities,
generalizations, and applications,” Cubo Matemática Educacional, vol. 5, no. 3, pp. 63–90, 2003.
6 E. Neuman and J. Sándor, “On the Schwab-Borchardt mean,” Mathematica Pannonica, vol. 14, no. 2,
pp. 253–266, 2003.
7 E. Neuman and J. Sándor, “On the Schwab-Borchardt mean. II,” Mathematica Pannonica, vol. 17, no. 1,
pp. 49–59, 2006.
8 W.-S. Cheung and F. Qi, “Logarithmic convexity of the one-parameter mean values,” Taiwanese Journal
of Mathematics, vol. 11, no. 1, pp. 231–237, 2007.
9 N.-G. Zheng, Z.-H. Zhang, and X.-M. Zhang, “Schur-convexity of two types of one-parameter mean
values in n variables,” Journal of Inequalities and Applications, vol. 2007, Article ID 78175, 10 pages,
2007.
10 M.-K. Wang, Y.-F. Qiu, and Y.-M. Chu, “An optimal double inequality among the one-parameter,
arithmetic and harmonic means,” Revue d’Analyse Numérique et de Théorie de l’Approximation, vol. 39,
no. 2, pp. 169–175, 2010.
11 Y.-M. Chu, C. Zong, and G.-D. Wang, “Optimal convex combination bounds of Seiffert and geometric
means for the arithmetic mean,” Journal of Mathematical Inequalities, vol. 5, no. 3, pp. 429–434, 2011.
12 Y.-M. Chu, M.-K. Wang, and W.-M. Gong, “Two sharp double inequalities for Seiffert mean,” Journal
of Inequalities and Applications, vol. 2011, article 44, 2011.
13 Y.-M. Chu, M.-K. Wang, S.-L. Qiu, and Y.-F. Qiu, “Sharp generalized Seiffert mean bounds for Toader
mean,” Abstract and Applied Analysis, vol. 2011, Article ID 605259, 8 pages, 2011.
14 S.-W. Hou and Y.-M. Chu, “Optimal convex combination bounds of root-square and harmonic rootsquare means for Seiffert mean,” International Journal of Mathematical Analysis, vol. 5, no. 39, pp. 1897–
1904, 2011.
15 M.-K. Wang, Y.-F. Qiu, and Y.-M. Chu, “Sharp bounds for Seiffert means in terms of Lehmer means,”
Journal of Mathematical Inequalities, vol. 4, no. 4, pp. 581–586, 2010.
16 Y.-M. Chu, M.-K. Wang, and Y.-F. Qiu, “An optimal double inequality between power-type Heron
and Seiffert means,” Journal of Inequalities and Applications, vol. 2010, Article ID 146945, 11 pages, 2010.
17 H.-N. Hu, C. Zong, and Y.-M. Chu, “Optimal one-parameter Gini mean bounds for the first and
second Seiffert means,” Pacific Journal of Applied Mathematics, vol. 4, no. 1, pp. 27–33, 2011.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014