Refinements of Inequalities among Difference of Means

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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,[email protected] 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 2ab

ab, N1a, b

√ a√

b 2

2

Aa, b Ga, b

2 ,

1.2

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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:

M_SAa, b Sa, b−Aa, b, MSN2a, b Sa, b−N2a, b, M_SN₃a, b Sa, b−N3a, b, MSN1a, b Sa, b−N1a, b, MSGa, b Sa, b−Ga, b, M_SHa, b Sa, b−Ha, b, MAN2a, b Aa, b−N2a, b,

MAGa, b Aa, b−Ga, b, M_AHa, b Aa, b−Ha, b, MN2N1a, b N2a, b−N1a, b,

M_N₂_Ga, 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,∞.

Further, using Theorem A, Taneja proved several chains of inequalities; they are refinements of inequalities in1.1.

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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 1

3MSHa, b−MSAa, b, D_AH−SHa, b 1

2MAHa, b−1

3MSHa, b, D_SG−AHa, b MSGa, b−MAHa, b, DAG−SGa, b MAGa, b−1

2MSGa, b, DN2N1−AHa, b MN2N1a, b− 1

8MAHa,b, DN2G−N2N1a, b 1

3MN2Ga, b−MN2N1a, b, D_AG−N₂Ga, b 1

4MAGa, b− 1

3MN2Ga, b, DAN2−AGa, b MAN2a, b−1

4MAGa, b, DSN2−SAa, b 4

5MSN2a, b−MSAa, b, DAN2−SN2a, b 4MAN2a, b−4

5MSN2a, b, DSN1−SHa, b 2MSN1a, b−MSHa, b, D_SG−SN₁a, b 3

2MSGa, b−2MSN1a, b,

1.10

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DSN3−SAa, b 3

4MSN3a, b−MSAa, b, DSN1−SN3a, b 2

3MSN1a, 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, . . . , x_n∈Rⁿandy y1, . . . , y_n∈Rⁿ. ix is said to be majorized by y in symbols x ≺ yif _k

i1x_i ≤ _k

i1y_i for k 1,2, . . . , n−1 and_n

i1xi _n

i1yi, 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, . . . , x_n∈Rⁿandy y1, . . . , y_n∈Rⁿ.

i Ω⊆Rⁿ is called a geometrically convex set ifx₁^αy₁^β, . . . , x^α_ny_n^β∈ 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 for everyn×npermutation matrixP.

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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 → Rbe 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 2

a²b²

2a²b²ab≥2

a²b²

ab²≥2ab², 2.7

so the right-hand inequality in2.2holds.

The inequality in2.3is equivalent to 2a²b²−ab

2a²b² ≥ a−b²

4ab². 2.8

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Since

2a²b²−ab

2a²b² 2

a²b²

−ab² 2a²b²

2a²b² ab a−b²

2a²b² ab

2a²b²,

2.9

so it is suﬃcient prove that

2

a²b²

ab

2a²b²≤4ab², 2.10

that is,

ab

2a²b²≤2

a²b²4ab

, 2.11

and, from the left-hand inequalities in2.2, we have ab

2a²b²≤2

a²b²

≤2

a²b²4ab

, 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

1−2t

16t²2t−1² 1/816t−7² 7/8

162t−1⁴ ≥0, 2.15

and the left-hand inequality in2.14reduces to

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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

D_SH−SAa, b 1

3M_SHa, b−M_SAa, b ab

2 − 2ab

3ab−2 3

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−SAa, b

∂a −b∂DSH−SAa, b

∂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 thatD_SH−SAis Schur-geometrically convex inR².

2For

DAH−SHa, b 1

2MAHa, b−1

3MSHa, b ab

4 − ab

3ab−1 3

a²b²

2 . 3.5

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To prove that the functionDAH−SHis Schur-geometrically convex inR²it is enough to notice thatD_AH−SHa, b 1/2D_SH−SAa, b.

3For

D_SG−AHa, b M_SGa, b−M_AHa, b

a²b²

2 −

ab−ab 2 2ab

ab, 3.6 we have

∂D_SG−AHa, b

∂a a

2a²b²− b 2√

ab −1

2 2b² ab²,

∂D_SG−AHa, b

∂b b

2a²b²− a 2√

ab −1

2 2a² ab²,

3.7

and then

Λ: lna−lnb

a∂D_SH−SAa, b

∂a −b∂D_SH−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 D_SH−SA is Schur- geometrically convex inR².

4For

D_AG−SGa, b M_AGa, b−1

2M_SGa, b 1 2

⎛

⎝ab− ab−

a²b² 2

⎞

⎠, 3.9

we have

∂D_AG−SGa, b

∂a 1

2

1− b 2√

ab− a

2a²b²

,

∂D_AG−SGa, b

∂b 1

2

1− a 2√

ab− b

2a²b²

,

3.10

and then

Λ: lna−lnb

a∂DSH−SAa, b

∂a −b∂DSH−SAa, b

∂b

a−blna−lnb

1− ab 2a²b²

.

3.11

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By2.2we infer that

1− ab

2a²b² ≥0, 3.12

soΛ≥0. ByLemma 2.4, we get thatD_AG−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

,

∂D_N₂_N₁_−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∂D_N₂_N₁_−AHa, b

∂a −b∂D_N₂_N₁_−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 thatD_N₂_N₁_−AHis Schur-geometrically convex inR².

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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 ,

∂D_N₂_G−N₂_N₁a, 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∂D_N₂_G−N₂_N₁a, b

∂a −b∂D_N₂_G−N₂_N₁a, 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 thatD_N₂_G−N₂_N₁ 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 12

2ab,

∂D_AG−N2Ga, b

∂b 1

8 a

24√ ab −

√ab 12√

2b −

√a√ b 12

2ab,

3.21

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and then

Λ lna−lnb

a∂D_AG−N₂_Ga, b

∂a −b∂D_AG−N₂_Ga, b

∂b

lna−lnb

⎛

⎜⎝a−b

8 −

√ab√ a−√

b 12√

2 −a−b√

a√ b 12

2ab

⎞

⎟⎠

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 thatD_AG−N₂_G is Schur-geometrically convex inR².

8In order to prove that the functionDAN2−AGa, bis Schur-geometrically convex in R²it is enough to notice that

D_AN₂_−AGa, b M_AN₂a, b−1

4M_AGa, b 3D_AG−N₂_Ga, b. 3.23

9For

D_SN₂_−SAa, b 4

5M_SN₂a, b−M_SAa, b ab

2 −1 5

a²b² 2 −1

5 √

a

b

2ab,

3.24

we have

∂DSN2−SAa, b

∂a 1

2 − a

5

2a²b²−1 5

ab 2a −

√a√ b 5

2ab,

∂D_SN2−SAa, b

∂b 1

2 − b

5

2a²b²−1 5

ab 2b −

√a√ b 5

2ab,

3.25

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and then

Λ lna−lnb

∂DSN2−SAa, b

∂a − ∂DSN2−SAa, b

∂b

lna−lnb

⎛

⎜⎝a−b

2 − a²−b² 5

2a²b²−1 5

⎛

⎝

aab

2 −

bab 2

⎞

⎠− √

a√ b

a−b 5

2ab

⎞

⎟⎠

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 functionD_SN₂_−SAa, bis Schur-geometrically convex inR². 10One can easily check that

D_AN_A_N₂_−SN₂a, b 4D_SN₂_−SAa, b, 3.28

and, consequently, the functionD_AN₂_−SN₂is Schur-geometrically convex inR². 11To prove that the function

D_SN₁_−SHa, b 2M_SN₁a, b−M_SHa, b

a²b²

2 −ab

2 −

ab 2ab

ab 3.29

is Schur-geometrically convex inR²it is enough to notice that

D_SN₁_−SHa, b D_SG−AHa, b. 3.30

12For

DSG−SN1a, b 3

2MSGa, b−2MSN1a, b 1

2

⎛

⎝ab− ab−

a²b² 2

⎞

⎠,

3.31

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we have

∂D_SG−SN₁a, b

∂a 1

2

1− b 2√

ab− a

2a²b²

,

∂D_SG−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 1

2DAG−SGa, b, 3.34

which means that the functionDSN3−SAis Schur-geometrically convex inR².

14 To prove that the function D_SN₁_−SN₃ is Schur-geometrically convex in R² it is enough to notice that

D_SN₁_−SN₃a, b 1

6D_AG−SGa, b. 3.35

The proof of Theorem I is complete.

4. Applications

Applying Theorem I,Lemma 2.6, andDefinition 2.2one can easily prove the following.

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Theorem II. Let 0< a≤b. 1/2≤t≤1 or 0≤t≤1/2,ua^tb^1−tandvb^ta^1−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−1

2MSGu, 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

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 EducationKM201011417013.

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