Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 315697, 15 pages
doi:10.1155/2012/315697
Research Article
Refinements of Inequalities among
Difference of Means
Huan-Nan Shi, Da-Mao Li, and Jian Zhang
Department of Electronic Information, Teacher’s College, Beijing Union University, Beijing 100011, China
Correspondence should be addressed to Huan-Nan Shi, shihuannan@yahoo.com.cn
Received 21 June 2012; Accepted 10 September 2012
Academic Editor: Janusz Matkowski
Copyright q 2012 Huan-Nan Shi 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.
In this paper, for the difference of famous means discussed by Taneja in 2005, we study the Schurgeometric convexity in 0, ∞ × 0, ∞ of the difference between them. Moreover some inequalities
related to the difference of those means are obtained.
1. Introduction
In 2005, Taneja 1 proved the following chain of inequalities for the binary means for a, b ∈
R2 0, ∞ × 0, ∞:
Ha, b ≤ Ga, b ≤ N1 a, b ≤ N3 a, b ≤ N2 a, b ≤ Aa, b ≤ Sa, b,
1.1
where
ab
,
2
Ga, b ab,
Aa, b Ha, b √
N1 a, b a
2
2ab
,
ab
√ 2
b
Aa, b Ga, b
,
2
1.2
2
International Journal of Mathematics and Mathematical Sciences
√
ab b 2Aa, b Ga, b
,
3
3
⎞
√
√ ⎛
a b ⎝ a b⎠
,
N2 a, b 2
2
N3 a, b a
Sa, b 1.3
a2 b2
.
2
The means A, G, H, S, N1 and N3 are called, respectively, the arithmetic mean, the geometric
mean, the harmonic mean, the root-square mean, the square-root mean, and Heron’s mean.
The N2 one can be found in Taneja 2, 3.
Furthermore Taneja considered the following difference of means:
MSA a, b Sa, b − Aa, b,
MSN2 a, b Sa, b − N2 a, b,
MSN3 a, b Sa, b − N3 a, b,
MSN1 a, b Sa, b − N1 a, b,
MSG a, b Sa, b − Ga, b,
MSH a, b Sa, b − Ha, b,
1.4
MAN2 a, b Aa, b − N2 a, b,
MAG a, b Aa, b − Ga, b,
MAH a, b Aa, b − Ha, b,
MN2 N1 a, b N2 a, b − N1 a, b,
MN2 G a, b N2 a, b − Ga, b
and established the following.
Theorem A. The difference of means given by 1.4 is nonnegative and convex in R2 0, ∞ ×
0, ∞.
Further, using Theorem A, Taneja proved several chains of inequalities; they are
refinements of inequalities in 1.1.
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3
Theorem B. The following inequalities among the mean differences hold:
MSA a, b ≤
1
1
1
MSH a, b ≤ MAH a, b ≤ MSG a, b ≤ MAG a, b,
3
2
2
1.5
1
1
1
MAH a, b ≤ MN2 N1 a, b ≤ MN2 G a, b ≤ MAG a, b ≤ MAN2 a, b,
8
3
4
MSA a, b ≤
4
MSN2 a, b ≤ 4MAN2 a, b,
5
1.7
3
MSG a, b,
2
1.8
3
2
MSN3 a, b ≤ MSN1 a, b.
4
3
1.9
MSH a, b ≤ 2MSN1 a, b ≤
MSA a, b ≤
1.6
For the difference of means given by 1.4, we study the Schur-geometric convexity of
difference between these differences in order to further improve the inequalities in 1.1. The
main result of this paper reads as follows.
Theorem I. The following differences are Schur-geometrically convex in R2 0, ∞ × 0, ∞:
DSH−SA a, b DAH−SH a, b 1
MSH a, b − MSA a, b,
3
1
1
MAH a, b − MSH a, b,
2
3
DSG−AH a, b MSG a, b − MAH a, b,
1
DAG−SG a, b MAG a, b − MSG a, b,
2
1
DN2 N1 −AH a, b MN2 N1 a, b − MAH a, b,
8
DN2 G−N2 N1 a, b 1
MN2 G a, b − MN2 N1 a, b,
3
1
1
DAG−N2 G a, b MAG a, b − MN2 G a, b,
4
3
1
DAN2 −AG a, b MAN2 a, b − MAG a, b,
4
DSN2 −SA a, b 4
MSN2 a, b − MSA a, b,
5
4
DAN2 −SN2 a, b 4MAN2 a, b − MSN2 a, b,
5
DSN1 −SH a, b 2MSN1 a, b − MSH a, b,
DSG−SN1 a, b 3
MSG a, b − 2MSN1 a, b,
2
1.10
4
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DSN3 −SA a, b DSN1 −SN3 a, b 3
MSN3 a, b − MSA a, b,
4
1.11
2
3
MSN1 a, b − MSN3 a, b.
3
4
The proof of this theorem will be given in Section 3. Applying this result, in Section 4,
we prove some inequalities related to the considered differences of means. Obtained
inequalities are refinements of inequalities 1.5–1.9.
2. Definitions and Auxiliary Lemmas
The Schur-convex function was introduced by Schur in 1923, and it has many important
applications in analytic inequalities, linear regression, graphs and matrices, combinatorial
optimization, information-theoretic topics, Gamma functions, stochastic orderings, reliability,
and other related fields cf. 4–14.
In 2003, Zhang first proposed concepts of “Schur-geometrically convex function”
which is extension of “Schur-convex function” and established corresponding decision
theorem 15. Since then, Schur-geometric convexity has evoked the interest of many
researchers and numerous applications and extensions have appeared in the literature cf.
16–19.
In order to prove the main result of this paper we need the following definitions and
auxiliary lemmas.
Definition 2.1 see 4, 20. Let x x1 , . . . , xn ∈ Rn and y y1 , . . . , yn ∈ Rn .
i x is said to be majorized by y in symbols x ≺ y if ki1 xi ≤ ki1 yi for k 1, 2, . . . , n − 1 and ni1 xi ni1 yi , where x1 ≥ · · · ≥ xn and y1 ≥ · · · ≥ yn are
rearrangements of x and y in a descending order.
ii Ω ⊆ Rn is called a convex set if αx1 βy1 , . . . , αxn βyn ∈ Ω for every x and y ∈ Ω,
where α and β ∈ 0, 1 with α β 1.
iii Let Ω ⊆ Rn . The function ϕ: Ω → R is said to be a Schur-convex function on Ω if
x ≺ y on Ω implies ϕx ≤ ϕy. ϕ is said to be a Schur-concave function on Ω if and
only if −ϕ is Schur-convex.
Definition 2.2 see 15. Let x x1 , . . . , xn ∈ Rn and y y1 , . . . , yn ∈ Rn .
β
β
i Ω ⊆ Rn is called a geometrically convex set if x1α y1 , . . . , xnα yn ∈ Ω for all x,y ∈ Ω
and α,β ∈ 0, 1 such that α β 1.
ii Let Ω ⊆ Rn . The function ϕ: Ω → R is said to be Schur-geometrically convex
function on Ω if ln x1 , . . . , ln xn ≺ ln y1 , . . . , ln yn on Ω implies ϕx ≤ ϕy. The
function ϕ is said to be a Schur-geometrically concave on Ω if and only if −ϕ is
Schur-geometrically convex.
Definition 2.3 see 4, 20. i The set Ω ⊆ Rn is called symmetric set, if x ∈ Ω implies P x ∈ Ω
for every n × n permutation matrix P .
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5
ii The function ϕ : Ω → R is called symmetric if, for every permutation matrix P ,
ϕP x ϕx for all x ∈ Ω.
Lemma 2.4 see 15. Let Ω ⊆ Rn be a symmetric and geometrically convex set with a nonempty
interior Ω0 . Let ϕ : Ω → R be continuous on Ω and differentiable in Ω0 . If ϕ is symmetric on Ω and
ln x1 − ln x2 x1
∂ϕ
∂ϕ
− x2
∂x1
∂x2
≥0
≤ 0
2.1
holds for any x x1 , . . . , xn ∈ Ω0 , then ϕ is a Schur-geometrically convex Schur-geometrically
concave function.
Lemma 2.5. For a, b ∈ R2 0, ∞ × 0, ∞ one has
1≥ ab
2a2
b2 ≥
2ab
1
,
2 a b2
ab
3
ab
≤ ,
−
2a2 b2 a b2 4
√
√
√
3
ab
5
ab
a b
≥
≥ .
√ √
2 √ 2 √a √b
2 a b 4 a b2
2.2
2.3
2.4
Proof. It is easy to see that the left-hand inequality in 2.2 is equivalent to a − b2 ≥ 0, and
the right-hand inequality in 2.2 is equivalent to
2a2 b2 − a b a b2 − 4ab
,
≤
2a b2
2a2 b2 2.5
that is,
a − b2
a − b2
.
≤
2a2 b2 2a2 b2 a b 2a b2
2.6
Indeed, from the left-hand inequality in 2.2 we have
2 a2 b2 2a2 b2 a b ≥ 2 a2 b2 a b2 ≥ 2a b2 ,
2.7
so the right-hand inequality in 2.2 holds.
The inequality in 2.3 is equivalent to
2a2 b2 − a b
a − b2
.
≥
4a b2
2a2 b2 2.8
6
International Journal of Mathematics and Mathematical Sciences
Since
2 a2 b2 − a b2
2a2 b2 − a b
2a2 b2 2a2 b2 2a2 b2 a b
2.9
a − b2
,
2a2 b2 a b 2a2 b2 so it is sufficient prove that
2 a2 b2 a b 2a2 b2 ≤ 4a b2 ,
2.10
a b 2a2 b2 ≤ 2 a2 b2 4ab ,
2.11
that is,
and, from the left-hand inequalities in 2.2, we have
a b 2a2 b2 ≤ 2 a2 b2 ≤ 2 a2 b2 4ab ,
2.12
so the inequality in 2.3 holds.
Notice that the functions in√the inequalities 2.4
√ are homogeneous. So, without loss of
√
generality, we may assume a b 1, and set t ab. Then 0 < t ≤ 1/4 and 2.4 reduces
to
3
≥
2
√
t2
1 − 2t
1
5
√ √
≥ .
√
2
2 1 − 2t 4 1 − 2t2
2.13
Squaring every side in the above inequalities yields
1
25
t4
9 1 − 2t
5t2
≥
1≥
.
4
2
2 − 4t
16 1 − 2t4 21 − 2t2
2.14
Reducing to common denominator and rearranging, the right-hand inequality in 2.14
reduces to
1 − 2t 16t2 2t − 12 1/816t − 72 7/8
162t − 14
and the left-hand inequality in 2.14 reduces to
≥ 0,
2.15
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7
21 − 2t2 2 − 51 − 2t
1 2t
−
≤ 0,
21 − 2t
2
2.16
so two inequalities in 2.4 hold.
Lemma 2.6 see 16. Let a ≤ b, ut ta 1 − tb, vt tb 1 − ta. If 1/2 ≤ t2 ≤ t1 ≤ 1 or
0 ≤ t1 ≤ t2 ≤ 1/2, then
ab ab
,
2
2
2.17
≺ ut2 , vt2 ≺ ut1 , vt1 ≺ a, b.
3. Proof of Main Result
Proof of Theorem I. Let a, b ∈ R2 .
1 For
2ab
2
1
ab
−
−
DSH−SA a, b MSH a, b − MSA a, b 3
2
3a b 3
a2 b2
,
2
3.1
we have
2b2
a
∂DSH−SA a, b 1
2
−
− ,
2
∂a
2 3a b
3 2a2 b2 3.2
2a2
b
∂DSH−SA a, b 1
2
−
− ,
∂b
2 3a b2 3 2a2 b2 whence
∂DSH−SA a, b
∂DSH−SA a, b
−b
∂a
∂b
1
2ab
ab
2
.
a − bln a − ln b
− 2 3a b2 3 2a2 b2 Λ : ln a − ln b a
3.3
From 2.3 we have
1
2ab
ab
2
− ≥ 0,
2 3a b2 3 2a2 b2 3.4
which implies Λ ≥ 0 and, by Lemma 2.4, it follows that DSH−SA is Schur-geometrically convex
in R2 .
2 For
ab
1
1
1
ab
−
−
DAH−SH a, b MAH a, b − MSH a, b 2
3
4
3a b 3
a2 b2
.
2
3.5
8
International Journal of Mathematics and Mathematical Sciences
To prove that the function DAH−SH is Schur-geometrically convex in R2 it is enough to
notice that DAH−SH a, b 1/2DSH−SA a, b.
3 For
DSG−AH a, b MSG a, b − MAH a, b 2ab
a2 b2 ab
− ab −
,
2
2
ab
3.6
we have
∂DSG−AH a, b
1
a
2b2
b
,
− √ − ∂a
2a2 b2 2 ab 2 a b2
∂DSG−AH a, b
1
b
2a2
a
,
− √ − ∂b
2a2 b2 2 ab 2 a b2
3.7
and then
∂DSH−SA a, b
∂DSH−SA a, b
−b
∂a
∂b
ab
2ab
1
a − bln a − ln b .
− −
2a2 b2 2 a b2
Λ : ln a − ln b a
3.8
From 2.2 we have Λ ≥ 0, so by Lemma 2.4, it follows that DSH−SA is Schurgeometrically convex in R2 .
4 For
⎛
⎞
a2 b2 ⎠
1
1⎝
a b − ab −
,
DAG−SG a, b MAG a, b − MSG a, b 2
2
2
3.9
we have
∂DAG−SG a, b 1
∂a
2
∂DAG−SG a, b 1
∂b
2
a
b
1− √ − 2 ab
2a2 b2 b
a
1− √ − 2 ab
2a2 b2 ,
3.10
,
and then
∂DSH−SA a, b
∂DSH−SA a, b
−b
∂a
∂b
ab
.
a − bln a − ln b 1 − 2a2 b2 Λ : ln a − ln b a
3.11
International Journal of Mathematics and Mathematical Sciences
9
By 2.2 we infer that
1− ab
2a2 b2 ≥ 0,
3.12
so Λ ≥ 0. By Lemma 2.4, we get that DAG−SG is Schur-geometrically convex in R2 .
5 For
1
DN2 N1 −AH a, b MN2 N1 a, b − MAH a, b
8
⎞
√
√ ⎛
2ab
a b ⎝ a b⎠ 1
1 ab
1
− a b −
−
,
ab −
2
2
4
2
8
2
ab
3.13
we have
√ b
ab
2
1 1
b
2b2
1
,
−
− − √ −
4 4 ab 8 2 a b2
√
√ ∂DN2 N1 −AH a, b
1
ab 1
a b
ab
√
∂b
2
4
2
2
4 b
1 1
a
2a2
1
,
−
− − √ −
4 4 ab 8 2 a b2
∂DN2 N1 −AH a, b
1
√
∂a
4 a
ab 1
2
4
√
a
2
−1/2
3.14
−1/2
and then
∂DN2 N1 −AH a, b
∂DN2 N1 −AH a, b
−b
∂a
∂b
⎞
⎛
√
√
√
ab ⎟
ab
a b
5
1
⎜
− −
a − bln a − ln b⎝ √ √
⎠.
√ √ √
4
4
2 ab
a b2
2 a b
Λ ln a − ln b a
3.15
From 2.4 we have
√
√
√
ab
5
ab
a b
− −
≥ 0,
√ √ √
√ √
2 a b 4 a b2
2 a b
so Λ ≥ 0; it follows that DN2 N1 −AH is Schur-geometrically convex in R2 .
3.16
10
International Journal of Mathematics and Mathematical Sciences
6 For
1
MN2 G a, b − MN2 N1 a, b
3
⎞
√
√ ⎛
√
ab
a b ⎝ a b⎠
ab 2
−
,
4
6
3
2
2
DN2 G−N2 N1 a, b 3.17
we have
∂DN2 G−N2 N1 a, b 1
b
1
√ − √
∂a
4 12 ab 6 a
∂DN2 G−N2 N1 a, b 1
1
a
√ − √
∂b
4 12 ab 6 b
ab 1
−
2
6
ab 1
−
2
6
√
√ a b
ab
2
2
√
a
2
√ b
ab
2
−1/2
,
3.18
−1/2
,
and then
Λ ln a − ln b a
⎛
∂DN2 G−N2 N1 a, b
∂DN2 G−N2 N1 a, b
−b
∂a
∂b
√ √
√ √
a
b
−
b
a
ab
a− b ab
⎜1
ln a − ln b⎝ a − b −
−
4
6
2
12
2
⎞
−1/2
⎟
⎠
3.19
⎞
√
√
ab
a b ⎟
1
⎜3
a − bln a − ln b⎝ − √ √
−√ √
⎠.
√
6
2
2 ab
2 a b
⎛
√
By 2.4 we infer that Λ ≥ 0, which proves that DN2 G−N2 N1 is Schur-geometrically convex in
R2 .
7 For
1
1
MAG a, b − MN2 G a, b
4
3
⎞
√
√ ⎛
ab
1
a b ⎝ a b⎠
1
,
ab −
8
12
3
2
2
DAG−N2 G a, b 3.20
we have
√
√
√
∂DAG−N2 G a, b 1
a b
b
ab
,
√ − √ − ∂a
8 24 ab 12 2a 12 2a b
√
√
√
∂DAG−N2 G a, b 1
a
ab
a b
√ − √ − ,
∂b
8 24 ab 12 2b 12 2a b
3.21
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11
and then
∂DAG−N2 G a, b
∂DAG−N2 G a, b
−b
∂a
∂b
⎛
√
√ ⎞
√
√
√
a
b
a
−
b
a
b
−
b
a
⎟
⎜a − b
ln a − ln b⎝
−
−
√
⎠
8
12 2
12 2a b
Λ ln a − ln b a
3.22
⎞⎞
√
√
ab
a b ⎟⎟
2⎜
a − bln a − ln b ⎜
⎠⎠.
⎝1 − ⎝ √ √
√ √ √
8
3
2 ab
2 a b
⎛
⎛
√
From 2.4 we have Λ ≥ 0, and, consequently, by Lemma 2.4, we obtain that DAG−N2 G is
Schur-geometrically convex in R2 .
8 In order to prove that the function DAN2 −AG a, b is Schur-geometrically convex in
2
R it is enough to notice that
1
DAN2 −AG a, b MAN2 a, b − MAG a, b 3DAG−N2 G a, b.
4
3.23
4
MSN2 a, b − MSA a, b
5
a b 1 a2 b2 1 √
−
−
a b
2a b,
2
5
2
5
3.24
9 For
DSN2 −SA a, b we have
∂DSN2 −SA a, b 1
a
1
− −
∂a
2 5 2a2 b2 5
∂DSN2 −SA a, b 1
b
1
− −
∂b
2 5 2a2 b2 5
√
√
ab
a b
− ,
2a
5 2a b
√
√
ab
a b
− ,
2b
5 2a b
3.25
12
International Journal of Mathematics and Mathematical Sciences
and then
∂DSN2 −SA a, b ∂DSN2 −SA a, b
−
∂a
∂b
⎛
⎛
Λ ln a − ln b
a −b
aa b
1
⎜a − b
ln a − ln b⎝
− −
− ⎝
2
2
2
2
5 2a b 5
2
a − bln a − ln b
√
5 2
2
⎞
ba b ⎠
−
2
√
⎞
√ a b a − b
⎟
⎠
5 2a b
√ √
√
ab
a b
5
ab
.
−√
√ − √
√ −√
2
a b
ab
a2 b2
3.26
From 2.2 and 2.4 we obtain that
√
√
√
√
5
5
ab
a b
ab
3
≥ √ − 2 − √ 0,
−√
√ − √
√ −√
2
2
2
2
2
a b
ab
a b
3.27
so Λ ≥ 0, which proves that the function DSN2 −SA a, b is Schur-geometrically convex in R2 .
10 One can easily check that
DANA N2 −SN2 a, b 4DSN2 −SA a, b,
3.28
and, consequently, the function DAN2 −SN2 is Schur-geometrically convex in R2 .
11 To prove that the function
DSN1 −SH a, b 2MSN1 a, b − MSH a, b a2 b2 a b 2ab
−
− ab 2
2
ab
3.29
is Schur-geometrically convex in R2 it is enough to notice that
DSN1 −SH a, b DSG−AH a, b.
3.30
12 For
3
MSG a, b − 2MSN1 a, b
2
⎛
⎞
2
2
1
a b ⎠
⎝a b − ab −
,
2
2
DSG−SN1 a, b 3.31
International Journal of Mathematics and Mathematical Sciences
13
we have
∂DSG−SN1 a, b 1
∂a
2
∂DSG−SN1 a, b 1
∂b
2
a
b
1− √ − 2 ab
2a2 b2 b
a
1− √ − 2 ab
2a2 b2 ,
3.32
,
and then
∂DSG−SN1 a, b
∂DSG−SN1 a, b
−b
∂a
∂b
ab
a − bln a − ln b
.
1− 2
2a2 b2 Λ ln a − ln b a
3.33
By the inequality 2.2 we get that Λ ≥ 0, which proves that DSG−SN1 is Schurgeometrically convex in R2 .
13 It is easy to check that
DSN3 −SA a, b 1
DAG−SG a, b,
2
3.34
which means that the function DSN3 −SA is Schur-geometrically convex in R2 .
14 To prove that the function DSN1 −SN3 is Schur-geometrically convex in R2 it is
enough to notice that
DSN1 −SN3 a, b 1
DAG−SG a, b.
6
3.35
The proof of Theorem I is complete.
4. Applications
Applying Theorem I, Lemma 2.6, and Definition 2.2 one can easily prove the following.
14
International Journal of Mathematics and Mathematical Sciences
Theorem II. Let 0 < a ≤ b. 1/2 ≤ t ≤ 1 or 0 ≤ t ≤ 1/2, u at b1−t and v bt a1−t . Then
MSA a, b ≤
1
MSH a, b −
3
1
MSH u, v − MSA u, v
3
≤
1
MAH a, b −
2
1
1
MAH u, v − MSH u, v
2
3
≤
1
≤ MSG a, b −
2
1
1
MSG u, v − MAH u, v
2
2
1
≤ MSG a, b
2
1
≤ MAG a, b − MAG u, v − MSG u, v
2
≤
1
MN2 G a, b −
3
1
≤ MAG a, b −
4
MSA a, b ≤
4
MSN2 a, b −
5
4.1
≤ MN2 N1 a, b
1
MN2 G u, v − MN2 N1 u, v
3
≤
1
1
MAG u, v − MN2 G u, v
4
3
1
≤ MAN2 a, b − MAN2 u, v − MAG u, v
4
1
MAH a, b
2
≤ MAG a, b,
1
1
MAH a, b ≤ MN2 N1 a, b − MN2 N1 u, v − MAH u, v
8
8
≤
1
MSH a, b
3
1
MN2 G a, b
3
1
≤ MAG a, b
4
4.2
≤ MAN2 a, b,
4
4
MSN2 u, v − MSN2 u, v
5
5
≤
4
≤ 4MAN2 a, b − 4MAN2 u, v − MSN2 u, v
5
4
MSN2 a, b
5
4.3
≤ 4MAN2 a, b,
MSH a, b ≤ 2MSN1 a, b − 2MSN1 u, v − MSH u, v ≤ 2MSN1 a, b
≤
MSA a, b ≤
3
MSG a, b −
2
3
3
MSG u, v − MSG u, v
2
2
≤
3
MSG a, b,
2
3
MSN3 a, b −
4
3
MSN3 u, v − MSA u, v
4
≤
3
MSN3 a, b
4
2
≤ MSN1 a, b −
3
2
3
MSN1 u, v − MSN3 u, v
3
4
2
≤ MSN1 a, b.
3
4.4
4.5
Remark 4.1. Equation 4.1, 4.2, 4.3, 4.4, and 4.5 are a refinement of 1.5, 1.6, 1.7,
1.8, and 1.9, respectively.
Acknowledgments
The authors are grateful to the referees for their helpful comments and suggestions. The
first author was supported in part by the Scientific Research Common Program of Beijing
Municipal Commission of Education KM201011417013.
International Journal of Mathematics and Mathematical Sciences
15
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