Volume 2012, Article ID 315697,15pages 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
Copyrightq2012 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 Schur-geometric convexity in0,∞×0,∞of the difference between them. Moreover some inequalities related to the difference of those means are obtained.
1. Introduction
In 2005, Taneja1proved the following chain of inequalities for the binary means fora, b∈
R2
0,∞×0,∞:
Ha, b≤Ga, b≤N1a, b≤N3a, b≤N2a, b≤Aa, b≤Sa, b, 1.1
where
Aa, b ab
2 ,
Ga, b ab,
Ha, b a2abb,
N1a, b
√
a√b
2 2
Aa, b Ga, b
2 ,
N3a, b a
√
abb
3
2Aa, b Ga, b
3 ,
N2a, b
√
a√b
2
⎛ ⎝
ab
2 ⎞ ⎠,
Sa, b
a2b2 2 .
1.3
The meansA,G,H,S,N1andN3are called, respectively, the arithmetic mean, the geometric mean, the harmonic mean, the root-square mean, the square-root mean, and Heron’s mean. TheN2one can be found in Taneja2,3.
Furthermore Taneja considered the following difference of means:
MSAa, b Sa, b−Aa, b,
MSN2a, b Sa, b−N2a, b,
MSN3a, b Sa, b−N3a, b,
MSN1a, b Sa, b−N1a, b,
MSGa, b Sa, b−Ga, b,
MSHa, b Sa, b−Ha, b,
MAN2a, b Aa, b−N2a, b,
MAGa, b Aa, b−Ga, b,
MAHa, b Aa, b−Ha, b,
MN2N1a, b N2a, b−N1a, b,
MN2Ga, b N2a, b−Ga, b
1.4
and established the following.
Theorem A. The difference of means given by1.4is nonnegative and convex inR2
0,∞×
0,∞.
Theorem B. The following inequalities among the mean differences hold:
MSAa, b≤ 1
3MSHa, b≤ 1
2MAHa, b≤ 1
2MSGa, b≤MAGa, b, 1.5
1
8MAHa, b≤MN2N1a, b≤ 1
3MN2Ga, b≤ 1
4MAGa, b≤MAN2a, b, 1.6
MSAa, b≤ 4
5MSN2a, b≤4MAN2a, b, 1.7
MSHa, b≤2MSN1a, b≤ 3
2MSGa, b, 1.8
MSAa, b≤ 3
4MSN3a, b≤ 2
3MSN1a, b. 1.9
For the difference of means given by1.4, we study the Schur-geometric convexity of difference between these differences in order to further improve the inequalities in1.1. The main result of this paper reads as follows.
Theorem I. The following differences are Schur-geometrically convex in R2
0,∞×0,∞:
DSH−SAa, b 13MSHa, b−MSAa, b,
DAH−SHa, b 1
2MAHa, b− 1
3MSHa, b,
DSG−AHa, b MSGa, b−MAHa, b,
DAG−SGa, b MAGa, b−12MSGa, b,
DN2N1−AHa, b MN2N1a, b−
1
8MAHa,b,
DN2G−N2N1a, b 13MN2Ga, b−MN2N1a, b,
DAG−N2Ga, b 1
4MAGa, b− 1
3MN2Ga, b,
DAN2−AGa, b MAN2a, b−1
4MAGa, b,
DSN2−SAa, b 45MSN2a, b−MSAa, b,
DAN2−SN2a, b 4MAN2a, b−4
5MSN2a, b,
DSN1−SHa, b 2MSN1a, b−MSHa, b,
DSG−SN1a, b 32MSGa, b−2MSN1a, b,
DSN3−SAa, b 34MSN3a, b−MSAa, b,
DSN1−SN3a, b 23MSN1a, b−
3
4MSN3a, b.
1.11
The proof of this theorem will be given inSection 3. Applying this result, inSection 4, we prove some inequalities related to the considered differences of means. Obtained inequalities are refinements of inequalities1.5–1.9.
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 fieldscf.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 literaturecf.
16–19.
In order to prove the main result of this paper we need the following definitions and auxiliary lemmas.
Definition 2.1see4,20. Letx x1, . . . , xn∈Rnandy y1, . . . , yn∈Rn.
ix is said to be majorized by y in symbols x ≺ yif ki1xi ≤ ki1yi for k
1,2, . . . , n−1 andni1xi ni1yi, wherex1 ≥ · · · ≥xnandy1 ≥ · · · ≥ynare rearrangements ofx and y in a descending order.
ii Ω⊆Rnis called a convex set ifαx1βy1, . . . , αxnβyn∈Ωfor everyx and y∈Ω, whereαandβ∈0,1withαβ1.
iiiLetΩ⊆ Rn. The functionϕ:Ω → Ris said to be a Schur-convex function onΩif
x≺y onΩimpliesϕx≤ϕy. ϕis said to be a Schur-concave function onΩif and only if−ϕis Schur-convex.
Definition 2.2see15. Letx x1, . . . , xn∈Rnandy y1, . . . , yn∈Rn.
i Ω⊆Rn is called a geometrically convex set ifx1αy1β, . . . , xαnynβ∈ Ωfor allx,y∈ Ω andα,β∈0,1such thatαβ1.
iiLet Ω ⊆ Rn. The function ϕ:Ω → R is said to be Schur-geometrically convex
function onΩiflnx1, . . . ,lnxn≺lny1, . . . ,lnynonΩimpliesϕx≤ϕy. The function ϕ is said to be a Schur-geometrically concave on Ωif and only if −ϕ is Schur-geometrically convex.
Definition 2.3see4,20. iThe setΩ⊆Rnis called symmetric set, ifx∈ΩimpliesPx∈Ω
iiThe functionϕ : Ω → Ris called symmetric if, for every permutation matrixP,
ϕPx ϕxfor allx∈Ω.
Lemma 2.4see15. LetΩ ⊆Rn be a symmetric and geometrically convex set with a nonempty
interiorΩ0. Letϕ:Ω → R
be continuous onΩand differentiable inΩ0. Ifϕis symmetric onΩand
lnx1−lnx2
x1
∂ϕ ∂x1 −x2
∂ϕ ∂x2
≥0 ≤0 2.1
holds for anyx x1, . . . , xn ∈ Ω0, thenϕis a Schur-geometrically convexSchur-geometrically
concavefunction.
Lemma 2.5. Fora, b∈R2 0,∞×0,∞one has
1≥ ab 2a2b2≥
1 2
2ab
ab2, 2.2
ab
2a2b2−
ab
ab2 ≤
3
4, 2.3
3 2 ≥
√
ab
√
2√a√b
√
a√b
√
2√ab ≥ 5 4
ab
ab2. 2.4
Proof. It is easy to see that the left-hand inequality in2.2is equivalent toa−b2 ≥0, and
the right-hand inequality in2.2is equivalent to
2a2b2−ab
2a2b2 ≤
ab2−4ab
2ab2 , 2.5
that is,
a−b2
2a2b2 2a2b2ab ≤
a−b2
2ab2. 2.6
Indeed, from the left-hand inequality in2.2we have
2a2b2
2a2b2ab≥2a2b2 ab2≥2ab2, 2.7
so the right-hand inequality in2.2holds. The inequality in2.3is equivalent to
2a2b2−ab
2a2b2 ≥
a−b2
Since
2a2b2−ab
2a2b2
2a2b2−ab2
2a2b22a2b2 ab
a−b2
2a2b2 ab2a2b2,
2.9
so it is sufficient prove that
2a2b2 ab
2a2b2≤4ab2, 2.10
that is,
ab2a2b2≤2a2b24ab, 2.11
and, from the left-hand inequalities in2.2, we have
ab2a2b2≤2a2b2≤2a2b24ab, 2.12
so the inequality in2.3holds.
Notice that the functions in the inequalities2.4are homogeneous. So, without loss of generality, we may assume√a√b1, and sett√ab. Then 0< t≤1/4 and2.4reduces to
3 2 ≥
√
1−2t
√
2
1
√
2√1−2t ≥ 5 4
t2
1−2t2. 2.13
Squaring every side in the above inequalities yields
9 4 ≥
1−2t
2
1
2−4t1≥ 25 16
t4
1−2t4
5t2
21−2t2. 2.14
Reducing to common denominator and rearranging, the right-hand inequality in 2.14
reduces to
1−2t16t22t−12 1/816t−72 7/8
162t−14 ≥0,
2.15
21−2t22−51−2t 21−2t −
12t
2 ≤0, 2.16
so two inequalities in2.4hold.
Lemma 2.6see16. Leta≤b, ut ta 1−tb, vt tb 1−ta. If 1/2≤t2≤t1 ≤1 or
0≤t1≤t2≤1/2, then ab
2 ,
ab
2
≺ut2, vt2≺ut1, vt1≺a, b. 2.17
3. Proof of Main Result
Proof of Theorem I. Leta, b∈R2
.
1For
DSH−SAa, b 13MSHa, b−MSAa, b a2b−32aabb−23
a2b2
2 , 3.1
we have
∂DSH−SAa, b
∂a
1 2−
2b2 3ab2 −
2 3
a
2a2b2,
∂DSH−SAa, b
∂b
1 2−
2a2 3ab2 −
2 3
b
2a2b2,
3.2
whence
Λ: lna−lnb
a∂DSH−∂aSAa, b−b∂DSH−∂bSAa, b
a−blna−lnb
1 2
2ab 3ab2 −
2 3
ab
2a2b2
.
3.3
From2.3we have
1 2
2ab 3ab2 −
2 3
ab
2a2b2 ≥0, 3.4
which impliesΛ≥0 and, byLemma 2.4, it follows thatDSH−SAis Schur-geometrically convex inR2
.
2For
DAH−SHa, b 12MAHa, b−13MSHa, b a4b−3aabb−13
a2b2
To prove that the functionDAH−SHis Schur-geometrically convex inR2it is enough to
notice thatDAH−SHa, b 1/2DSH−SAa, b.
3For
DSG−AHa, b MSGa, b−MAHa, b
a2b2
2 −
ab−ab
2
2ab
ab, 3.6
we have
∂DSG−AHa, b
∂a
a
2a2b2−
b
2√ab − 1 2
2b2
ab2,
∂DSG−AHa, b
∂b
b
2a2b2−
a
2√ab − 1 2
2a2
ab2,
3.7
and then
Λ: lna−lnb
a∂DSH−SAa, b ∂a −b
∂DSH−SAa, b
∂b
a−blna−lnb
ab
2a2b2− 1 2 −
2ab
ab2
.
3.8
From 2.2 we have Λ ≥ 0, so by Lemma 2.4, it follows that DSH−SA is Schur-geometrically convex inR2
.
4For
DAG−SGa, b MAGa, b−12MSGa, b 12 ⎛
⎝ab−ab−
a2b2 2
⎞
⎠, 3.9
we have
∂DAG−SGa, b
∂a
1 2
1− b
2√ab−
a
2a2b2
,
∂DAG−SGa, b
∂b
1 2
1− a
2√ab−
b
2a2b2
,
3.10
and then
Λ: lna−lnb
a∂DSH−∂aSAa, b−b∂DSH−∂bSAa, b
a−blna−lnb
1− ab 2a2b2
.
By2.2we infer that
1− ab
2a2b2 ≥0, 3.12
soΛ≥0. ByLemma 2.4, we get thatDAG−SGis Schur-geometrically convex inR2
.
5For
DN2N1−AHa, b MN2N1a, b−
1
8MAHa, b
√
a√b
2
⎛ ⎝
ab
2 ⎞ ⎠− 1
4ab− 1 2
ab−1
8 ab
2 − 2ab
ab
,
3.13
we have
∂DN2N1−AHa, b
∂a
1 4√a
ab
2
1 4
√
a√b
2
ab
2 −1/2
−1
4 −
b
4√ab− 1 8
1 2−
2b2
ab2
,
∂DN2N1−AHa, b
∂b
1 4√b
ab
2
1 4
√
a√b
2
ab
2 −1/2
−1
4 −
a
4√ab− 1 8
1 2−
2a2
ab2
,
3.14
and then
Λ lna−lnb
a∂DN2N1−AHa, b
∂a −b
∂DN2N1−AHa, b
∂b
1
4a−blna−lnb ⎛ ⎜ ⎝
√
ab
√
2√a√b
√
a√b
√
2√ab− 5 4 −
ab
ab2
⎞ ⎟ ⎠.
3.15
From2.4we have
√
ab
√
2√a√b
√
a√b
√
2√ab− 5 4−
ab
ab2 ≥0, 3.16
soΛ≥0; it follows thatDN2N1−AHis Schur-geometrically convex inR2
6For
DN2G−N2N1a, b
1
3MN2Ga, b−MN2N1a, b
ab
4 √ ab 6 − 2 3 √
a√b
2
⎛ ⎝
ab
2 ⎞
⎠, 3.17
we have
∂DN2G−N2N1a, b
∂a
1 4
b
12√ab − 1 6√a
ab
2 − 1 6
√
a√b
2
ab
2 −1/2
,
∂DN2G−N2N1a, b
∂b
1 4
a
12√ab− 1 6√b
ab
2 − 1 6
√
a√b
2
ab
2 −1/2
,
3.18
and then
Λ lna−lnb
a∂DN2G−N2N1a, b
∂a −b
∂DN2G−N2N1a, b
∂b
lna−lnb ⎛ ⎜ ⎝1
4a−b−
√
a−√b
6
ab
2 −
a−b√a√b
12
ab
2 −1/2 ⎞ ⎟ ⎠ 1
6a−blna−lnb ⎛ ⎜ ⎝3
2−
√
ab
√
2√a√b−
√a√
b
√
2√ab ⎞ ⎟ ⎠.
3.19
By2.4we infer thatΛ ≥0, which proves thatDN2G−N2N1 is Schur-geometrically convex in
R2
.
7For
DAG−N2Ga, b
1
4MAGa, b− 1
3MN2Ga, b
ab
8
1 12
ab− 1
3
√
a√b
2
⎛ ⎝
ab
2 ⎞
⎠,
3.20
we have
∂DAG−N2Ga, b
∂a
1 8
b
24√ab −
√
ab
12√2a−
√
a√b
122ab,
∂DAG−N2Ga, b
∂b
1 8
a
24√ab −
√
ab
12√2b −
√a√
b
122ab,
and then
Λ lna−lnb
a∂DAG−N2Ga, b
∂a −b
∂DAG−N2Ga, b
∂b
lna−lnb ⎛ ⎜ ⎝a−8b−
√
ab√a−√b
12√2 −
a−b√a√b
122ab ⎞ ⎟ ⎠
a−blna−lnb
8
⎛ ⎜ ⎝1−2
3 ⎛ ⎜ ⎝
√
ab
√
2√a√b
√
a√b
√
2√ab ⎞ ⎟ ⎠
⎞ ⎟ ⎠.
3.22
From 2.4 we have Λ ≥ 0, and, consequently, by Lemma 2.4, we obtain thatDAG−N2G is Schur-geometrically convex inR2
.
8In order to prove that the functionDAN2−AGa, bis Schur-geometrically convex in
R2
it is enough to notice that
DAN2−AGa, b MAN2a, b−
1
4MAGa, b 3DAG−N2Ga, b. 3.23
9For
DSN2−SAa, b 45MSN2a, b−MSAa, b
ab
2 − 1 5
a2b2
2 −
1 5
√
ab2ab,
3.24
we have
∂DSN2−SAa, b
∂a
1 2 −
a
52a2b2− 1 5
ab
2a −
√
a√b
52ab,
∂DSN2−SAa, b
∂b
1 2 −
b
52a2b2− 1 5
ab
2b −
√a√
b
52ab,
and then
Λ lna−lnb
∂D
SN2−SAa, b
∂a −
∂DSN2−SAa, b
∂b
lna−lnb
⎛ ⎜
⎝a−2b− a2−b2
52a2b2− 1 5
⎛ ⎝
aab
2 −
bab
2 ⎞ ⎠−
√
a√ba−b
52ab ⎞ ⎟ ⎠
a−blna−lnb
5√2
5
√
2−
ab
√
a2b2 −
√
ab
√
a√b−
√
a√b
√
ab
.
3.26
From2.2and2.4we obtain that
5
√
2 −
ab
√
a2b2 −
√
ab
√
a√b−
√
a√b
√
ab ≥
5
√
2−
√
2−√3
2 0, 3.27
soΛ≥0, which proves that the functionDSN2−SAa, bis Schur-geometrically convex inR2.
10One can easily check that
DANAN2−SN2a, b 4DSN2−SAa, b, 3.28
and, consequently, the functionDAN2−SN2is Schur-geometrically convex inR2
.
11To prove that the function
DSN1−SHa, b 2MSN1a, b−MSHa, b
a2b2
2 −
ab
2 −
aba2abb 3.29
is Schur-geometrically convex inR2
it is enough to notice that
DSN1−SHa, b DSG−AHa, b. 3.30
12For
DSG−SN1a, b 32MSGa, b−2MSN1a, b
1
2 ⎛
⎝ab−ab−
a2b2 2
⎞
⎠,
we have
∂DSG−SN1a, b
∂a
1 2
1− b
2√ab−
a
2a2b2
,
∂DSG−SN1a, b
∂b
1 2
1− a
2√ab−
b
2a2b2
,
3.32
and then
Λ lna−lnb
a∂DSG−SN1a, b ∂a −b
∂DSG−SN1a, b
∂b
a−blna−lnb
2
1− ab 2a2b2
.
3.33
By the inequality 2.2 we get that Λ ≥ 0, which proves that DSG−SN1 is
Schur-geometrically convex inR2
.
13It is easy to check that
DSN3−SAa, b 12DAG−SGa, b, 3.34
which means that the functionDSN3−SAis Schur-geometrically convex inR2.
14 To prove that the function DSN1−SN3 is Schur-geometrically convex in R2
it is
enough to notice that
DSN1−SN3a, b
1
6DAG−SGa, b. 3.35
The proof of Theorem I is complete.
4. Applications
Theorem II. Let 0< a≤b. 1/2≤t≤1 or 0≤t≤1/2,uatb1−tandvbta1−t. Then
MSAa, b≤ 1
3MSHa, b−
1
3MSHu, v−MSAu, v
≤ 1
3MSHa, b
≤ 1
2MAHa, b−
1
2MAHu, v− 1
3MSHu, v
≤ 1
2MAHa, b
≤ 1
2MSGa, b−
1
2MSGu, v− 1
2MAHu, v
≤ 1
2MSGa, b
≤MAGa, b−
MAGu, v−12MSGu, v
≤MAGa, b,
4.1
1
8MAHa, b≤MN2N1a, b−
MN2N1u, v−1
8MAHu, v
≤MN2N1a, b
≤ 1
3MN2Ga, b−
1
3MN2Gu, v−MN2N1u, v
≤ 1
3MN2Ga, b
≤ 1
4MAGa, b−
1
4MAGu, v− 1
3MN2Gu, v
≤ 1
4MAGa, b
≤MAN2a, b−
MAN2u, v−
1
4MAGu, v
≤MAN2a, b,
4.2
MSAa, b≤ 4
5MSN2a, b−
4
5MSN2u, v− 4
5MSN2u, v
≤ 4
5MSN2a, b
≤4MAN2a, b−
4MAN2u, v−
4
5MSN2u, v
≤4MAN2a, b,
4.3
MSHa, b≤2MSN1a, b−2MSN1u, v−MSHu, v≤2MSN1a, b
≤ 3
2MSGa, b−
3
2MSGu, v− 3
2MSGu, v
≤ 3
2MSGa, b,
4.4
MSAa, b≤ 3
4MSN3a, b−
3
4MSN3u, v−MSAu, v
≤ 3
4MSN3a, b
≤ 2
3MSN1a, b−
2
3MSN1u, v− 3
4MSN3u, v
≤ 2
3MSN1a, b.
4.5
Remark 4.1. Equation4.1,4.2,4.3,4.4, and4.5are a refinement of1.5,1.6,1.7,
1.8, and1.9, respectively.
Acknowledgments
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