Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 146945, 11 pages
doi:10.1155/2010/146945
Research Article
An Optimal Double Inequality between
Power-Type Heron and Seiffert Means
Yu-Ming Chu,1 Miao-Kun Wang,2 and Ye-Fang Qiu2
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 29 August 2010; Accepted 16 November 2010
Academic Editor: Alexander I. Domoshnitsky
Copyright q 2010 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 k ∈ 0, ∞, the power-type Heron mean Hk a, b and the Seiffert mean T a, b of two positive
√
real numbers a and b are defined by Hk a, b ak abk/2 bk /31/k , k / 0; Hk a, b ab,
k 0 and T a, b a − b/2 arctana − b/a b, a /
b; T a, b a, a b, respectively. In this
paper, we find the greatest value p and the least value q such that the double inequality Hp a, b <
T a, b < Hq a, b holds for all a, b > 0 with a /
b.
1. Introduction
For k ∈ 0, ∞, the power-type Heron mean Hk a, b and the Seiffert mean T a, b of two
positive real numbers a and b are defined by
⎧
1/k
k/2
k
k
⎪
b
a
ab
⎪
⎨
, k/
0,
3
Hk a, b ⎪
⎪
⎩√
ab,
k 0,
⎧
a−b
⎪
⎨
, a/
b,
T a, b 2 arctana − b/a b
⎪
⎩a,
a b,
respectively.
1.1
1.2
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Journal of Inequalities and Applications
Recently, the means of two variables have been the subject of intensive research 1–
15. In particular, many remarkable inequalities for Hk a, b and T a, b can be found in the
literature 16–20.
It is well known that Hk a, b is continuous and strictly increasing with respect to
1/b−a
,
k ∈ 0, ∞ for fixed a, b > 0 with a /
b. Let√Aa, b a b/2, Ia, b 1/ebb /aa La, b b − a/log b − log a, Ga, b ab, and Ha, b 2ab/a b be the arithmetic,
identric, logarithmic, geometric, and harmonic means of two positive numbers a and b with
a
/ b, respectively. Then
min{a, b} < Ha, b < Ga, b < La, b < Ia, b < Aa, b < max{a, b}.
1.3
For p ∈ R, the power mean Mp a, b of order p of two positive numbers a and b is
defined by
⎧
p
p 1/p
⎪
⎪
⎨ a b
,
2
Mp a, b ⎪
⎪
⎩√ab,
p/
0,
1.4
p 0.
The main properties for power mean are given in 21.
In 16, Jia and Cao presented the inequalities
H0 a, b Ga, b < La, b < Hp a, b < Mq a, b,
Aa, b < Hlog 3/ log 2 a, b
1.5
for all a, b > 0 with a / b, p ≥ 1/2, and q ≥ 2/3p.
Sándor 22 proved that
Ia, b > H1 a, b
1.6
for all a, b > 0 with a / b.
In 19, Seiffert established that
M1 a, b < T a, b < M2 a, b
1.7
for all a, b > 0 with a / b.
The purpose of this paper is to present the optimal upper and lower power-type Heron
mean bounds for the Seiffert mean T a, b. Our main result is the following Theorem 1.1.
Theorem 1.1. For all a, b > 0 with a / b, one has
Hlog 3/ logπ/2 a, b < T a, b < H5/2 a, b,
1.8
and Hlog 3/ logπ/2 a, b and H5/2 a, b are the best possible lower and upper power-type Heron mean
bounds for the Seiffert mean T a, b, respectively.
Journal of Inequalities and Applications
3
2. Lemmas
In order to prove our main result, Theorem 1.1, we need two lemmas which we present in
this section.
Lemma 2.1. If k log 3/ logπ/2 2.43 . . . and t > 1, then
− 24k − 2k 3k 43k 2tk8 48kk − 1k 33k − 2tk6
− k 4k 6k 84k − 7t8 < 0.
2.1
Proof. For t > 1, we clearly see that
− 24k − 2k 3k 43k 2tk8 48kk − 1k 3
× 3k − 2tk6 − k 4k 6k 84k − 7t8
< t8 −24k − 2k 3k 43k 2t2 48kk − 1k 3
2.2
×3k − 2t − k 4k 6k 84k − 7.
Let
ht −24k − 2k 3k 43k 2t2 48kk − 1k 3
× 3k − 2t − k 4k 6k 84k − 7.
2.3
Then
h1 68k4 − 281k3 − 1010k2 2072k 2496 −104.992 . . . < 0
2.4
and ht is strictly decreasing in 1, ∞ because of kk − 1k 33k − 2/k − 2k 3k 43k 2 < 1 for k log 3/ logπ/2.
Therefore, Lemma 2.1 follows from 2.2–2.4 together with the monotonicity of ht.
Lemma 2.2. If k log 3/ logπ/2 2.43 . . ., t ∈ 1, ∞, and gt −8t4k−4 8t4k−6 2 −
kt3k2 2kt3k − 2k 2t3k−2 2k − 4t3k−4 10 − kt3k−6 7 − 4kt2k2 24k − 1t2k − 24k 5t2k−2 24k−1t2k−4 7−4kt2k−6 10−ktk2 2k−4tk −2k2tk−2 2ktk−4 2−ktk−6 8t2 −8,
then there exists λ ∈ 1, ∞ such that gt > 0 for t ∈ 1, λ and gt < 0 for t ∈ λ, ∞.
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Journal of Inequalities and Applications
Proof. Let g1 t g t/t, g2 t t9−k g1 t, g3 t g2 t/2t, g4 t g3 t/2t, g5 t g4 t/kt,
g6 t g5 t/t, and g7 t t9−k g6 t. Then elaborated computations lead to
g1 0,
2.5
lim gt −∞,
2.6
t → ∞
g1 t −32k − 1t4k−6 162k − 3t4k−8 2 − k3k 2t3k
6k2 t3k−2 − 2k 23k − 2t3k−4 2k − 43k − 4t3k−6
3k − 210 − kt3k−8 2k 17 − 4kt2k 4k4k − 1t2k−2
−4k − 14k 5t2k−4 4k − 24k − 1t2k−6 27 − 4k
2.7
×k − 3t2k−8 k 210 − ktk 2kk − 4tk−2
−2k − 2k 2tk−4 2kk − 4tk−6 2 − kk − 6tk−8 16,
g1 1 0,
2.8
lim g1 t −∞,
2.9
t → ∞
g2 t −64k − 12k − 3t3k2 64k − 22k − 3t3k 3k2 − k
×3k 2t2k8 6k2 3k − 2t2k6 − 2k 23k − 23k − 4t2k4
6k − 2k − 43k − 4t2k2 3k − 210 − k3k − 8t2k
4kk 17 − 4ktk8 8kk − 14k − 1tk6 − 8k − 1
2.10
×k − 24k 5tk4 8k − 2k − 34k − 1tk2 4k − 4
×k − 37 − 4ktk kk 210 − kt8 2kk − 2k − 4t6
−2k − 2k − 4k 2t4 2kk − 4k − 6t2 2 − kk − 6k − 8,
g2 1 1445 − 2k > 0,
2.11
lim g2 t −∞,
2.12
t → ∞
g3 t −32k − 12k − 33k 2t3k 96kk − 22k − 3t3k−2
3k2 − kk 43k 2t2k6 6k2 k 33k − 2t2k4
−2k 22 3k − 23k − 4t2k2 6k 1k − 2k − 4
×3k − 4t2k 3kk − 210 − k3k − 8t2k−2 2kk 1
×7 − 4kk 8tk6 4kk − 1k 64k − 1tk4
k2
−4k − 1k − 2k 44k 5t
4k − 2k − 3k 2
×4k − 1tk 2kk − 4k − 37 − 4ktk−2 4kk 2
×10 − kt6 6kk − 2k − 4t4 − 4k − 2k − 4k 2t2
2kk − 4k − 6,
2.13
Journal of Inequalities and Applications
5
g3 1 725k − 25 − 2k > 0,
2.14
lim g3 t −∞,
2.15
t → ∞
g4 t −48kk − 12k − 33k 2t3k−2 48kk − 22k − 3
×3k − 2t3k−4 3k2 − kk 3k 43k 2t2k4
6k2 k 2k 33k − 2t2k2 − 2k 1k 22 3k − 2
×3k − 4t2k 6kk 1k − 2k − 43k − 4t2k−2
3kk − 1k − 210 − k3k − 8t2k−4 kk 1k 6
2.16
×7 − 4kk 8tk4 2kk − 1k 4k 64k − 1tk2
−2k − 1k − 2k 2k 44k 5tk 2kk − 2k − 3
×k 24k − 1tk−2 kk − 2k − 3k − 47 − 4ktk−4
12kk 210 − kt4 12kk − 2k − 4t2 − 4k − 2k − 4k 2,
g4 1 4 −318k3 885k2 − 210k − 72 304.99 . . . > 0,
lim g4 t −∞,
t → ∞
2.17
2.18
g5 t −48k − 12k − 33k − 23k 2t3k−4 48k − 22k − 3
×3k − 23k − 4t3k−6 62 − kk 2k 3k 43k 2
×t2k2 12kk 1k 2k 33k − 2t2k − 4k 1k 22
×3k − 23k − 4t2k−2 12k − 1k − 2k − 4k 13k − 4
×t2k−4 6k − 1k − 22 10 − k3k − 8t2k−6 k 1k 4
2.19
×k 6k 87 − 4ktk2 2k − 1k 2k 4k 6
×4k − 1tk − 2k − 1k − 2k 2k 44k 5tk−2
2k − 22 k − 3k 24k − 1tk−4 k − 2k − 3k − 42
×7 − 4ktk−6 48k 210 − kt2 24k − 2k − 4,
g5 1 4 −1038k3 3549k2 − 3360k 2196 323.50 . . . > 0,
lim g5 t −∞,
t → ∞
2.20
2.21
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Journal of Inequalities and Applications
g6 t −48k − 12k − 33k − 23k − 43k 2t3k−6 144k − 22
×2k − 33k − 23k − 4t3k−8 122 − kk 1k 2
×k 3k 43k 2t2k 24k2 k 1k 2k 3
×3k − 2t2k−2 − 8k − 1k 1k 22 3k − 23k − 4t2k−4
24k − 1k − 22 k − 4k 13k − 4t2k−6 12k − 1
×k − 22 k − 310 − k3k − 8t2k−8 k 1k 2
×k 4k 6k 87 − 4ktk 2kk − 1k 2k 4
×k 64k − 1tk−2 − 2k − 1k − 22 k 2k 4
×4k 5tk−4 2k − 22 k − 3k − 4k 24k − 1tk−6
k − 2k − 3k − 42 k − 67 − 4ktk−8 96k 210 − k,
g6 1 4 −3348k4 16233k3 − 30204k2 28092k − 6768 −2933.37 . . . < 0,
2.22
g7 t −144k − 1k − 22k − 33k − 23k − 43k 2t2k2
−144k − 22 2k − 33k − 23k − 48 − 3kt2k − 24kk − 2
×k 1k 2k 3k 43k 2tk8 48k2 k − 1k 1
×k 2k 33k − 2tk6 − 16k − 1k − 2k 1k 22
×3k − 23k − 4tk4 48k − 1k − 22 3 − k4 − kk 1
×3k − 4tk2 − 24k − 1k − 22 3 − k4 − k10 − k8 − 3k
k
2.23
×t − kk 1k 2k 4k 6k 84k − 7t 2k
8
×k − 1k − 2k 2k 4k 64k − 1t6 2k − 1
×k − 22 4 − kk 2k 44k 5t4 − 2k − 22 3 − k
×4 − k6 − kk 24k − 1t2 k − 23 − kk − 42
×6 − k4k − 78 − k.
From the expression of g7 t and Lemma 2.1, we get
g7 t < − 144k − 1k − 22k − 33k − 23k − 43k 2 48k − 1
× k − 22 3 − k4 − kk 13k − 4 2kk − 1k − 2k 2
× k 4k 64k − 1 2k − 1k − 22 4 − kk 2k 4
×4k 5 k − 23 − kk − 42 6 − k8 − k4k − 7 t2k2
kk 1k 2 −24k − 2k 3k 43k 2tk8 48k
×k − 1k 33k − 2tk6 − k 4k 6k 84k − 7t8
Journal of Inequalities and Applications
140k7 − 9353k6 52543k5 − 103636k4 51700k3 88448k2
7
−131968k 54016 t2k2 kk 1k 2
× −24k − 2k 3k 43k 2tk8 48kk − 1k 33k − 2tk6
−k 4k 6k 84k − 7t8
−20221.36 . . .t2k2 kk 1k 2
× −24k − 2k 3k 43k 2tk8 48kk − 1k 33k − 2tk6
−k 4k 6k 84k − 7t8
< 0.
2.24
From 2.24, we know that g6 t is strictly decreasing in 1, ∞. Then 2.22 implies that g5 t
is strictly decreasing in 1, ∞.
From 2.20 and 2.21 together with the monotonicity of g5 t, we clearly see that
there exists λ1 ∈ 1, ∞ such that g4 t is strictly increasing in 1, λ1 and strictly decreasing in
λ1 , ∞.
Inequality 2.17 and 2.18 together with the piecewise monotonicity of g4 t imply
that there exists λ2 ∈ 1, ∞ such that g3 t is strictly increasing in 1, λ2 and strictly
decreasing in λ2 , ∞.
The piecewise monotonicity of g3 t together with 2.14 and 2.15 leads to the fact
that there exists λ3 ∈ 1, ∞ such that g2 t is strictly increasing in 1, λ3 and strictly
decreasing in λ3 , ∞.
From 2.11 and 2.12 together with the piecewise monotonicity of g2 t, we conclude
that there exists λ4 ∈ 1, ∞ such that g1 t is strictly increasing in 1, λ4 and strictly
decreasing in λ4 , ∞.
Equations 2.8 and 2.9 together with the piecewise monotonicity of g1 t imply that
there exists λ5 ∈ 1, ∞ such that gt is strictly increasing in 1, λ5 and strictly decreasing in
λ5 , ∞.
Therefore, Lemma 2.2 follows from 2.5 and 2.6 together with the piecewise
monotonicity of gt.
3. Proof of Theorem 1.1
Proof of Theorem 1.1. Withoutloss of generality, we assume that a > b. We first prove that
T a, b < H5/2 a, b. Let t 4 a/b > 1, then from 1.1 and 1.2 we have
log T a, b − log H5/2 a, b log
t10 t5 1
2
t4 − 1
.
4
− log
4
5
3
2 arctan t − 1 / t 1
3.1
8
Journal of Inequalities and Applications
Let
ft log
t10 t5 1
2
t4 − 1
log
.
−
5
3
2 arctan t4 − 1 / t4 1
3.2
Then simple computations lead to
limft 0,
t→1
2t3 2t6 t5 t 2
f t 4
f1 t,
t − 1 t10 t5 1 arctan t4 − 1 / t4 1
3.3
where f1 t arctant4 − 1/t4 1 − 2t4 − 1t10 t5 1/t8 12t6 t5 t 2. Note that
limf1 t 0,
t→1
f1 t
−
2 t2 1 t 12 t − 14
2
1 t8 2t6 t5 t 2
3.4
f t,
2 2
where
f2 t t18 2t17 4t16 6t15 − 8t12 6t11 21t10 28t9
21t8 6t7 − 8t6 6t3 4t2 2t 1 > 0
3.5
for t > 1.
Therefore, T a, b < H5/2 a, b follows from 3.1–3.5.
Next, we prove that T a, b > Hlog 3/ logπ/2 a, b. Let k log 3/ logπ/2 2.43 . . .
and t a/b > 1, then 1.1 and 1.2 lead to
log T a, b − log Hk a, b log
t2k tk 1
1
t2 − 1
log
.
−
3
2 arctant2 − 1/t2 1 k
3.6
Let
Ft log
t2k tk 1
t2 − 1
1
.
− log
2
2
3
2 arctant − 1/t 1 k
3.7
Then simple computations lead to
lim Ft lim Ft 0,
t→1
F t t2
− 1
t2k
t → ∞
2t2k−1 tk1 tk−1 2t
F1 t,
tk 1 arctant2 − 1/t2 1
3.8
3.9
Journal of Inequalities and Applications
9
where F1 t arctant2 − 1/t2 1 − 2t2 − 1t2k tk 1/t4 12t2k−2 tk tk−2 2. Note
that
lim F1 t 0,
t→1
lim F1 t t → ∞
π
− 1 < 0,
4
2t3
2 F2 t,
2
t4 1 2t2k−2 tk tk−2 2
F t 3.10
3.11
3.12
where
F2 t −8t4k−4 8t4k−6 2 − kt3k2 2kt3k − 2k 2t3k−2
2k − 4t3k−4 10 − kt3k−6 7 − 4kt2k2
24k − 1t2k − 24k 5t2k−2 24k − 1t2k−4
3.13
7 − 4kt2k−6 10 − ktk2 2k − 4tk
− 2k 2tk−2 2ktk−4 2 − ktk−6 8t2 − 8.
From 3.12 and 3.13 together with Lemma 2.2, we clearly see that there exists λ ∈
1, ∞ such that F1 t is strictly increasing in 1, λ and strictly decreasing in λ, ∞.
Equations 3.9–3.11 and the piecewise monotonicity of F1 t imply that there exists
μ ∈ 1, ∞ such that Ft is strictly increasing in 1, μ and strictly decreasing in μ, ∞. Then
from 3.8 we get
Ft > 0
3.14
for t > 1.
Therefore, T a, b > Hlog 3/ logπ/2 a, b follows from 3.6 and 3.7 together with
3.14.
At last, we prove that Hlog 3/ logπ/2 a, b and H5/2 a, b are the best possible lower and
upper power-type Heron mean bounds for the Seiffert mean T a, b, respectively.
For any 0 < ε < k log 3/ logπ/2 2.43 . . . and x > 0, from 1.1 and 1.2, one has
Jx
,
T 1, 1 x5/2−ε − H5/2−ε 1, 1 x5/2−ε 5/2−ε
3 2
arctanx/x 25/2−ε
lim
x → ∞
Hkε x, 1 π −1/kε π −1/k
·3
> ·3
1,
T x, 1
2
2
where Jx 3x5/2−ε − 1 x5/2−ε 1 x5/4−ε/2 12 arctanx/x 25/2−ε .
3.15
3.16
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Journal of Inequalities and Applications
Let x → 0, making use of Taylor extension, we get
5/2−ε
x x2 x3
3
Jx 3x
−
o x
−2
2
4
12
13 5
5 ε 2
15 3
− ε x
− ε
−
× 3
x o x2
4
2
8
4
4 2
1
ε5 − 2εx9/2−ε o x9/2−ε .
8
5/2−ε
5/2−ε
3.17
Equations 3.15 and 3.17 together with inequality 3.16 imply that for any 0 < ε <
log 3/ logπ/2, there exist δ δε > 0 and X Xε > 1 such that T 1, 1 x > H5/2−ε 1, 1 x for x ∈ 0, δ and Hlog 3/ logπ/2ε 1, x > T 1, x for x ∈ X, ∞.
Acknowledgments
This work was supported by the Natural Science Foundation of China under Grant no.
11071069, the Natural Science Foundation of Zhejiang Province under Grant no. Y7080106,
and the Innovation Team Foundation of the Department of Education of Zhejiang Province
under Grant no. T200924.
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