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Two Sharp Inequalities for Power Mean, Geometric Mean, and Harmonic Mean

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Chu Yu-Ming
Chu Yu-Ming
Chu Yu-Ming
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Xia Wei-Feng
Xia Wei-Feng
Xia Wei-Feng
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Abstract

For p. R, the power mean of order p of two positive numbers a and b is defined by M-p (a, b) = ((a(p) + b(p))/2)(1/p), p not equal 0, and M-p (a, b) = root ab, p = 0. In this paper, we establish two sharp inequalities as follows: (2/3)G(a, b) + (1/3)H(a, b) >= M-1/3(a, b) and (1/3)G(a, b) + (2/3)H(a, b) >= M-2/3(a, b) for all a, b > 0. Here G(a, b) = root ab and H(a, b) = 2ab/(a + b) denote the geometric mean and harmonic mean of a and b, respectively. Copyright (C) 2009 Y.-M. Chu and W.-F. Xia.
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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 741923, 6 pages
doi:10.1155/2009/741923
Research Article
Two Sharp Inequalities for Power Mean,
Geometric Mean, and Harmonic Mean
Yu-Ming Chu
1
and Wei-Feng Xia
2
1
Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2
School of Teacher Education, Huzhou Teachers College, Huzhou 313000, China
Correspondence should be addressed to Yu-Ming Chu, chuyuming2005@yahoo.com.cn
Received 23 July 2009; Accepted 30 October 2009
Recommended by Wing-Sum Cheung
For p ∈ R, the power mean of order p of two positive numbers a and b is defined by
M
p
a, ba
p
b
p
/2
1/p
,p
/
0, and M
p
a, b
ab, p 0. In this paper, we establish two
sharp inequalities as follows: 2/3Ga, b1/3Ha, b
M
−1/3
a, band 1/3Ga, b
2/3Ha, b
M
−2/3
a, b for all a, b > 0. Here Ga, b
√
ab and Ha, b2ab/a b denote
the geometric mean and harmonic mean of a and b, respectively.
Copyright q 2009 Y.-M. Chu and W.-F. Xia. 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.
1. Introduction
For p ∈ R, the power mean of order p of two positive numbers a and b is defined by
M
p
a, b
⎧
⎪
⎪
⎨
⎪
⎪
⎩
a
p
b
p
2
1/p
,p
/
0,
√
ab, p 0.
1.1
Recently, the power mean has been the subject of intensive research. In particular,
many remarkable inequalities for M
p
a, b can be found in literature 1–12. It is well known
that M
p
a, b is continuous and increasing with respect to p ∈ R for fixed a and b.Ifwe
denote by Aa, ba b/2,Ga, b
√
ab, and Ha, b2ab/a b the arithmetic mean,
geometric mean and harmonic mean of a and b, respectively, then
min
{
a, b
}
H
a, b
M
−1
a, b
G
a, b
M
0
a, b
A
a, b
M
1
a, b
max
{
a, b
}
.
1.2
2 Journal of Inequalities and Applications
In 13, Alzer and Janous established the following sharp double-inequality see also
14, page 350:
M
log 2/ log 3
a, b
2
3
A
a, b
1
3
G
a, b
M
2/3
a, b
1.3
for all a, b > 0.
In 15, Mao proved
M
1/3
a, b
1
3
A
a, b
2
3
G
a, b
M
1/2
a, b
1.4
for all a, b > 0, and M
1/3
a, b is the best possible lower power mean bound for the sum
1/3Aa, b2/3Ga, b.
The purpose of this paper is to answer the questions: what are the greatest values p and
q, and the least values r and s, such that M
p
a, b 2/3Ga, b1/3Ha, b M
r
a, b
and M
q
a, b 1/3Ga, b2/3Ha, b M
s
a, b for all a, b > 0?
2. Main Results
Theorem 2.1. 2/3Ga, b1/3Ha, b M
−1/3
a, b for all a, b > 0, equality holds if and
only if a b, and M
−1/3
a, b is the best possible lower power mean bound for the sum 2/3Ga, b
1/3Ha, b.
Proof. If a b, then we clearly see that 2/3Ga, b1/3Ha, bM
−1/3
a, ba.
If a
/
b and a/b t
6
, then simple computation leads to
2
3
G
a, b
1
3
H
a, b
− M
−1/3
a, b
b
2t
3
3
2t
6
3
1 t
6
−
8t
6
1 t
2
3
2bt
3
3
1 t
2
3
t
4
− t
2
1
×
t
2
1
3
t
4
− t
2
1
t
3
t
2
1
2
− 12t
3
t
4
− t
2
1
2bt
3
3
1 t
2
3
t
4
− t
2
1
×
t
10
2t
8
− 11t
7
t
6
14t
5
t
4
− 11t
3
2t
2
1
2bt
3
t − 1
4
3
1 t
2
3
t
4
− t
2
1
×
t
6
4t
5
12t
4
17t
3
12t
2
4t 1
> 0.
2.1
Next, we prove that M
−1/3
a, b is the best possible lower power mean bound for the
sum 2/3Ga, b1/3Ha, b.
Journal of Inequalities and Applications 3
For any 0 <ε<
1
3
and 0 <x<1, one has
M
−1/3ε
1 x
2
, 1
1/3−ε
−
2
3
G
1 x
2
, 1
1
3
H
1 x
2
, 1
1/3−ε
1
1 x
−2/32ε
2
−1
−
2
3
1 x
2
1 x
2
3
x
2
2x 2
1/3−ε
2
1 x
2/3−2ε
1
1 x
2/3−2ε
−
1 2x 4/3x
2
x
3
/3
1 x x
2
/2
1/3−ε
f
x
1
1 x
2/3−2ε
1 x x
2
/2
1/3−ε
,
2.2
where fx21 x
2/3−2ε
1 x x
2
/2
1/3−ε
− 1 1 x
2/3 −2ε
1 2x 4/3x
2
x
3
/3
1/3−ε
.
Let x → 0, then the Taylor expansion leads to
f
x
2
1
2 − 6ε
3
x −
1 − 3ε

1 6ε
9
x
2
o
x
2
×
1
1 − 3ε
3
x
1 − 3ε
2
18
x
2
o
x
2
− 2
1
1 − 3ε
3
x −
1 − 3ε

1 6ε
18
x
2
o
x
2
×
1
2 − 6ε
3
x −
2ε
1 − 3ε
3
x
2
o
x
2
2
1
1 − 3ε
x
1 − 3ε

1 − 9ε
6
x
2
o
x
2
− 2
1
1 − 3ε
x
1 − 3ε

1 − 10ε
6
x
2
o
x
2
ε
1 − 3ε
3
x
2
o
x
2
.
2.3
Equations 2.2 and 2.3 imply that for any 0 <ε<1/3 there exists 0 <δ δε < 1,
such that M
−1/3ε
1 x
2
, 1 > 2/3G1 x
2
, 11/3H1 x
2
, 1 for x ∈ 0,δ.
Remark 2.2. For any ε>0, one has
lim
t →∞
2
3
G
1,t
1
3
H
1,t
− M
−ε
1,t
lim
t →∞
2
3
√
t
2t
3
1 t
−
2t
ε
1 t
ε
1/ε
∞.
2.4
4 Journal of Inequalities and Applications
Therefore, M
0
a, bGa, b is the best possible upper power mean bound for the
sum 2/3Ga, b1/3Ha, b.
Theorem 2.3. 1/3Ga, b2/3Ha, b M
−2/3
a, b for all a, b > 0, equality holds if and
only if a b, and M
−2/3
a, b is the best possible lower power mean bound for the sum 1/3Ga, b
2/3Ha, b.
Proof. If a b, then we clearly see that 1/3Ga, b2/3Ha, bM
−2/3
a, ba.
If a
/
b and a/b t
6
, then elementary calculation yields
1
3
G
a, b
2
3
H
a, b
2
−
M
−2/3
a, b

2
b
2
⎡
⎣
t
3
3
4t
6
3
1 t
6
2
−
2t
4
1 t
4
3
⎤
⎦
b
2
t
6
9
1 t
6
2
1 t
4
3
t
4
1
3
t
6
4t
3
1
2
− 72t
6
t
6
1
2
b
2
t
6
9
1 t
6
2
1 t
4
3

t
24
8t
21
3t
20
18t
18
24t
17
3t
16
8t
15
54t
14
24t
13
2t
12
24t
11
54t
10
8t
9
3t
8
24t
7
18t
6
3t
4
8t
3
1
−
72t
18
144t
12
72t
6

b
2
t
6
9
1 t
6
2
1 t
4
3
t
24
8t
21
3t
20
− 54t
18
24t
17
3t
16
8t
15
54t
14
24t
13
− 142t
12
24t
11
54t
10
8t
9
3t
8
24t
7
− 54t
6
3t
4
8t
3
1
b
2
t
6
t − 1
4
9
1 t
6
2
1 t
4
3
t
20
4t
19
10t
18
28t
17
70t
16
148t
15
220t
14
268t
13
277t
12
240t
11
240t
10
240t
9
277t
8
268t
7
220t
6
148t
5
70t
4
28t
3
10t
2
4t 1
> 0.
2.5
Next, we prove that M
−2/3
a, b is the best possible lower power mean bound for the
sum 1/3Ga, b2/3Ha, b.
Journal of Inequalities and Applications 5
For any 0 <ε<2/3and0<x<1, one has
M
−2/3ε
1,
1 x
2
2/3−ε
−
1
3
G1,
1 x
2

2
3
H1,
1 x
2
2/3−ε
2
1 x
4−6ε/3
1
1 x
4−6ε/3
−
1 2x 7/6x
2
1/6x
3
2−3ε/3
1 x
1/2
x
2
2−3ε/3
f
x
1
1 x
4−6ε
/3
1 x
1/2
x
2
2−3ε/3
,
2.6
where fx21 x
4−6ε
/3
1 x x
2
/2
2−3ε
/3
− 1 2x 7/6x
2
1/6x
3
2−3ε
/3
1
1 x
4−6ε
/3
.
Let x → 0, then the Taylor expansion leads to
f
x
2
1
4 − 6ε
3
x
2 − 3ε

1 − 6ε
9
x
2
o
x
2
×
1
2 − 3ε
3
x
2 − 3ε
2
18
x
2
o
x
2
− 2
1
4 − 6ε
3
x
2 − 3ε

1 − 4ε
6
x
2
o
x
2
×
1
2 − 3ε
3
x
2 − 3ε

1 − 6ε
18
x
2
o
x
2
2
1
2 − 3ε
x
2 − 3ε

4 − 9ε
6
x
2
o
x
2
− 2
1
2 − 3ε
x
2 − 3ε

4 − 10ε
6
x
2
o
x
2
ε
2 − 3ε
3
x
2
o
x
2
.
2.7
Equations 2.6 and 2.7 imply that for any 0 <ε<2/3 there exists 0 <δ δε < 1,
such that
M
−2/3ε
1,
1 x
2
>
1/3
G
1,
1 x
2
2/3
H
1,
1 x
2
2.8
for x ∈ 0,δ.
Remark 2.4. For any ε>0, one has
lim
t →∞
1
3
G
1,t
2
3
H
1,t
− M
−ε
1,t
lim
t →∞
1
3
√
t
4t
3
1 t
−
2t
ε
1 t
ε
1/ε
∞.
2.9
6 Journal of Inequalities and Applications
Therefore, M
0
a, bGa, b is the best possible upper power mean bound for the
sum 1/3Ga, b2/3Ha, b.
Acknowledgments
This research is partly supported by N S Foundation of China under Grant 60850005 and the
N S Foundation of Zhejiang Province under Grants Y7080185 and Y607128.
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Thesis
  • Apr 2017
La diffusion Raman exaltée de surface (SERS) est utilisée pour la mise au point d’un biocapteur capable de détecter des pesticides dans l’eau, en se basant sur le suivi de l’activité enzymatique de l’Acétylcholinestérase (ACHE). Les nanoparticules d’or sont utilisées comme substrats SERS actifs. Le signal Raman exalté de l’analyte est optimisé en testant plusieurs types de nanoparticules.Le Raman SERS a permis la détection directe du Paraoxon (PO) et du carbaryl (CA) et la possibilité de suivi de l’activité de l’ACHE. En absence d'inhibiteurs, la molécule d’acétylcholine (ATC) est transformée en acide acétique et en choline par l’enzyme ACHE. La mesure de l’activité de l’ACHE repose sur le suivi des concentrations en ATC car sa transformation est inhibée en présence de pesticides. Il a été ainsi possible d’établir une relation linéaire entre la concentration de pesticides et l’exaltation du signal Raman de l’ATC non transformé. La méthode a permis la détection du PO et du CA, avec une limite de détection beaucoup plus faible que la détection directe. Ce biocapteur basé sur l’activité de l’ACHE a ensuite été utilisé pour l'évaluation d’autres polluants (inhibiteurs d’ACHE) comme les additifs contenus dans les plastiques notamment. Enfin, nous avons développé une seconde approche qui consistait à mesurer l’activité de l’ACHE en utilisant la diffusion dynamique de la lumière. En effet, nous avons montré que les paramètres physicochimiques (agrégation) des AuNPs en contact avec certaines molécules, sont fortement influencés par l’activité enzymatique de l’ACHE. C’est ce phénomène d’instabilité qui nous a permis de distinguer entre les deux cas : absence et présence de PO.
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Ostrowski type inequalities involving conformable fractional integrals
Article
Full-text available
  • Apr 2018
  • J INEQUAL APPL
In the article, we establish several Ostrowski type inequalities involving the conformable fractional integrals. As applications, we find new inequalities for the arithmetic and generalized logarithmic means.
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Landen inequalities for a class of hypergeometric functions with applications
Article
  • Apr 2018
  • MATH INEQUAL APPL
In this paper, we study a class of Gaussian hypergeometric function 2F1(a,b;(a+b+ 1)/2; x)(a, b > 0), and find the maximal regions of ab plane in the first quadrant where the well-known Landen identities for the complete elliptic integrals of the first kind turn on respective inequalities valid for each x ? (0, 1) . Besides, the generalized Grötzsch ring function with two parameters ?a,b(r) is introduced, and the analogs of duplication formula satisfied by Grötzsch ring function ? (r) for ?a,b(r), in the form of inequalities, will be derived.
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Optimal bounds for two Sándor-type means in terms of power means
Article
Full-text available
  • Feb 2016
  • J INEQUAL APPL
In the article, we prove that the double inequalities M α ( a , b ) < S Q A ( a , b ) < M β ( a , b ) and M λ ( a , b ) < S A Q ( a , b ) < M μ ( a , b ) hold for all a , b > 0 with a ≠ b if and only if α ≤ log 2 / [ 1 + log 2 − 2 log ( 1 + 2 ) ] = 1.5517 … , β ≥ 5 / 3 , λ ≤ 4 log 2 / [ 4 + 2 log 2 − π ] = 1.2351 … and μ ≥ 4 / 3 , where S Q A ( a , b ) = A ( a , b ) e Q ( a , b ) / M ( a , b ) − 1 and S A Q ( a , b ) = Q ( a , b ) e A ( a , b ) / T ( a , b ) − 1 are the Sándor-type means, A ( a , b ) = ( a + b ) / 2 , Q ( a , b ) = ( a 2 + b 2 ) / 2 , T ( a , b ) = ( a − b ) / [ 2 arctan ( ( a − b ) / ( a + b ) ) ] , and M ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / ( a + b ) ) ] are, respectively, the arithmetic, quadratic, second Seiffert, and Neuman-Sándor means. MSC: 26E60.
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On the optimal values for inequalities involving power means
Article
Full-text available
  • Jan 2004
Let Mn[r](a)betheM_n^{[r]} (a) be the r−thpowermeanof-th power mean of a, 0 <\alpha<\theta<\beta.Thenthelargestnumber. Then the largest number \lambda,satisfying, satisfying [M_n^{[\alpha]}]^{1-\lambda}[M_n^{[\beta]}]^{\lambda}\leq M_n^{[\theta]}(a),is, is \theta\geq{1+ (\beta-\theta)/[m(\theta-\alpha)]}-1,where, where m = \min\{[2 + (n - 2)t^{\beta}]/[2 + (n - 2)t^{\alpha], t\in R_{++}\}; the smallest numberλ \lambda, satisfying the reverse inequality, isλ=[θ(θ−α)]/[θ(θ−α)] \lambda= [\theta(\theta-\alpha)]/[\theta(\theta-\alpha)]. The method depends on the theory of majorization and analytic techniques. The optimality idea of establish-ing inequalities is displayed as many times as possible. As some applications, several inequalities (including improved Hardy's inequality) involving sums, integrals and ma-trices are obtained.
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An inequality for mixed power means
Article
  • Apr 1999
  • MATH INEQUAL APPL
In 1992 Frank Hollad [1] stated the following inequality (1) (A 1 A 2 A n )<SUP>1/n</SUP> greater than or equal to 1/n (G 1 G 2 ...G n ) where A(k), G(k), k = 1, 2,..., n are arithmetic and geometric means, respectively, of positive numbers a(1), a(2),,...,a(k). In 1994 Kiran Kedlaya [2] gave a combinatorial proof of (1). In 1995 Takashi Matsuda [3] gave another proof of(1). In 1996 B. Mend and J. Pecaric [5] proved the following generalization of inequality (1) involving power means: (2) if S > T then, m(r,s) (a) greater than or equal to m(s,r) (a), where m(r,s) (a) is defined by the following definition (1.2). In this article a more general inequality, which concern weighted power means, is proved.
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Generalization of the power means and their inequalities
Article
  • Nov 1991
In this paper we gave a generalization of power means which include positive nonlinear functionals. For these means we obtained generalizations of fundamental mean inequality, Hölder's and Minkowski's, and their converse inequalities.
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A Power Mean Inequality for the Gamma Function
Article
  • Dec 2000
A classical inequality for Euler’s gamma function states that for all and with . We prove the following extension of this result. Let be the weighted power mean of of order r. The inequality holds for all and with if and only if
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