# Refinements of Inequalities among Difference of Means

Authors: Huan-Nan Shi; Da-Mao Li; Jian Zhang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

In this paper, for the difference of famous means discussed by Taneja in 2005, we study the Schur-geometric convexity in (0, ∞) × (0, ∞) of the difference between them. Moreover some
inequalities related to the difference of those means are obtained.


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## Body

## 1. Introduction

In 2005, Taneja [1] proved the following chain of inequalities for the binary means for (a,b)∈R+2=(0,∞)×(0,∞):
(1.1)H(a,b)≤G(a,b)≤N1(a,b)≤N3(a,b)≤N2(a,b)≤A(a,b)≤S(a,b),
where
(1.3)A(a,b)=a+b2,G(a,b)=ab,H(a,b)=2aba+b,N1(a,b)=(a+b2)2=A(a,b)+G(a,b)2,N3(a,b)=a+ab+b3=2A(a,b)+G(a,b)3,N2(a,b)=(a+b2)(a+b2),S(a,b)=a2+b22.
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:(1.4)MSA(a,b)=S(a,b)-A(a,b),MSN2(a,b)=S(a,b)-N2(a,b),MSN3(a,b)=S(a,b)-N3(a,b),MSN1(a,b)=S(a,b)-N1(a,b),MSG(a,b)=S(a,b)-G(a,b),MSH(a,b)=S(a,b)-H(a,b),MAN2(a,b)=A(a,b)-N2(a,b),MAG(a,b)=A(a,b)-G(a,b),MAH(a,b)=A(a,b)-H(a,b),MN2N1(a,b)=N2(a,b)-N1(a,b),MN2G(a,b)=N2(a,b)-G(a,b)
and established the following.Theorem A.
The difference of means given by (1.4) is nonnegative and convex in R+2=(0,∞)×(0,∞).Further, using Theorem A, Taneja proved several chains of inequalities; they are refinements of inequalities in (1.1).Theorem B.
The following inequalities among the mean differences hold:(1.5)MSA(a,b)≤13MSH(a,b)≤12MAH(a,b)≤12MSG(a,b)≤MAG(a,b),(1.6)18MAH(a,b)≤MN2N1(a,b)≤13MN2G(a,b)≤14MAG(a,b)≤MAN2(a,b),(1.7)MSA(a,b)≤45MSN2(a,b)≤4MAN2(a,b),(1.8)MSH(a,b)≤2MSN1(a,b)≤32MSG(a,b),(1.9)MSA(a,b)≤34MSN3(a,b)≤23MSN1(a,b).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 inR+2=(0,∞)×(0,∞):
(1.11)DSH-SA(a,b)=13MSH(a,b)-MSA(a,b),DAH-SH(a,b)=12MAH(a,b)-13MSH(a,b),DSG-AH(a,b)=MSG(a,b)-MAH(a,b),DAG-SG(a,b)=MAG(a,b)-12MSG(a,b),DN2N1-AH(a,b)=MN2N1(a,b)-18MAH(a,b),DN2G-N2N1(a,b)=13MN2G(a,b)-MN2N1(a,b),DAG-N2G(a,b)=14MAG(a,b)-13MN2G(a,b),DAN2-AG(a,b)=MAN2(a,b)-14MAG(a,b),DSN2-SA(a,b)=45MSN2(a,b)-MSA(a,b),DAN2-SN2(a,b)=4MAN2(a,b)-45MSN2(a,b),DSN1-SH(a,b)=2MSN1(a,b)-MSH(a,b),DSG-SN1(a,b)=32MSG(a,b)-2MSN1(a,b),DSN3-SA(a,b)=34MSN3(a,b)-MSA(a,b),DSN1-SN3(a,b)=23MSN1(a,b)-34MSN3(a,b).The proof of this theorem will be given in Section3. 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]).
Letx=(x1,…,xn)∈ℝn and y=(y1,…,yn)∈ℝn. (i)
x is said to be majorized by y (in symbols x≺y) if ∑i=1k‍x[i]≤∑i=1k‍y[i] for k=1,2,…,n-1 and ∑i=1n‍xi=∑i=1n‍yi, where x[1]≥⋯≥x[n] and y[1]≥⋯≥y[n] 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)
LetΩ⊆ℝn. The function φ: Ω→ℝ is said to be a Schur-convex function on Ω if x≺y on Ω implies φ(x)≤φ(y).φ is said to be a Schur-concave function on Ω if and only if -φ is Schur-convex.Definition 2.2 (see [15]).
Letx=(x1,…,xn)∈ℝn and y=(y1,…,yn)∈ℝ+n. (i)
Ω⊆ℝ+n is called a geometrically convex set if (x1αy1β,…,xnαynβ)∈Ω for all x,y∈Ω and α,β∈[0,1] such that α+β=1.(ii)
LetΩ⊆ℝ+n. The function φ: Ω→ℝ+ is said to be Schur-geometrically convex function on Ω if (lnx1,…,lnxn)≺(lny1,…,lnyn) on Ω implies φ(x)≤φ(y). 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 setΩ⊆ℝn is called symmetric set, if x∈Ω implies Px∈Ω for every n×n permutation matrix P.
(ii) The functionφ:Ω→ℝ is called symmetric if, for every permutation matrix P, φ(Px)=φ(x) for all x∈Ω.Lemma 2.4 (see [15]).
LetΩ⊆ℝ+n 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
(2.1)(lnx1-lnx2)(x1∂φ∂x1-x2∂φ∂x2)≥0(≤0)
holds for any x=(x1,…,xn)∈Ω0, then φ is a Schur-geometrically convex (Schur-geometrically concave) function.Lemma 2.5.
For(a,b)∈R+2=(0,∞)×(0,∞) one has
(2.2)1≥a+b2(a2+b2)≥12+2ab(a+b)2,(2.3)a+b2(a2+b2)-ab(a+b)2≤34,(2.4)32≥a+b2(a+b)+a+b2a+b≥54+ab(a+b)2.Proof.
It is easy to see that the left-hand inequality in (2.2) is equivalent to (a-b)2≥0, and the right-hand inequality in (2.2) is equivalent to
(2.5)2(a2+b2)-(a+b)2(a2+b2)≤(a+b)2-4ab2(a+b)2,
that is,
(2.6)(a-b)22(a2+b2)+2(a2+b2)(a+b)≤(a-b)22(a+b)2.
Indeed, from the left-hand inequality in (2.2) we have
(2.7)2(a2+b2)+2(a2+b2)(a+b)≥2(a2+b2)+(a+b)2≥2(a+b)2,
so the right-hand inequality in (2.2) holds.
The inequality in (2.3) is equivalent to
(2.8)2(a2+b2)-(a+b)2(a2+b2)≥(a-b)24(a+b)2.
Since
(2.9)2(a2+b2)-(a+b)2(a2+b2)=2(a2+b2)-(a+b)22(a2+b2)(2(a2+b2)+(a+b))=(a-b)22(a2+b2)+(a+b)2(a2+b2),
so it is sufficient prove that
(2.10)2(a2+b2)+(a+b)2(a2+b2)≤4(a+b)2,
that is,
(2.11)(a+b)2(a2+b2)≤2(a2+b2+4ab),
and, from the left-hand inequalities in (2.2), we have
(2.12)(a+b)2(a2+b2)≤2(a2+b2)≤2(a2+b2+4ab),
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
(2.13)32≥1-2t2+121-2t≥54+t2(1-2t)2.
Squaring every side in the above inequalities yields
(2.14)94≥1-2t2+12-4t+1≥2516+t4(1-2t)4+5t22(1-2t)2.
Reducing to common denominator and rearranging, the right-hand inequality in (2.14) reduces to
(2.15)(1-2t)(16t2(2t-1)2+(1/8)(16t-7)2+(7/8))16(2t-1)4≥0,
and the left-hand inequality in (2.14) reduces to
(2.16)2(1-2t)2+2-5(1-2t)2(1-2t)=-1+2t2≤0,
so two inequalities in (2.4) hold.Lemma 2.6 (see [16]).
Leta≤b,u(t)=ta+(1-t)b,v(t)=tb+(1-t)a. If 1/2≤t2≤t1≤1 or 0≤t1≤t2≤1/2, then
(2.17)(a+b2,a+b2)≺(u(t2),v(t2))≺(u(t1),v(t1))≺(a,b).


## 3. Proof of Main Result

Proof of Theorem I.
Let(a,b)∈R+2.
(1) For(3.1)DSH-SA(a,b)=13MSH(a,b)-MSA(a,b)=a+b2-2ab3(a+b)-23a2+b22,
we have
(3.2)∂DSH-SA(a,b)∂a=12-2b23(a+b)2-23a2(a2+b2),∂DSH-SA(a,b)∂b=12-2a23(a+b)2-23b2(a2+b2),
whence
(3.3)Λ∶=(lna-lnb)(a∂DSH-SA(a,b)∂a-b∂DSH-SA(a,b)∂b)=(a-b)(lna-lnb)(12+2ab3(a+b)2-23a+b2(a2+b2)).
From (2.3) we have
(3.4)12+2ab3(a+b)2-23a+b2(a2+b2)≥0,
which implies Λ≥0 and, by Lemma 2.4, it follows that DSH-SA is Schur-geometrically convex in R+2.
(2) For(3.5)DAH-SH(a,b)=12MAH(a,b)-13MSH(a,b)=a+b4-ab3(a+b)-13a2+b22.
To prove that the functionDAH-SH is Schur-geometrically convex in R+2 it is enough to notice that DAH-SH(a,b)=(1/2)DSH-SA(a,b).
(3) For(3.6)DSG-AH(a,b)=MSG(a,b)-MAH(a,b)=a2+b22-ab-a+b2+2aba+b,
we have
(3.7)∂DSG-AH(a,b)∂a=a2(a2+b2)-b2ab-12+2b2(a+b)2,∂DSG-AH(a,b)∂b=b2(a2+b2)-a2ab-12+2a2(a+b)2,
and then
(3.8)Λ∶=(lna-lnb)(a∂DSH-SA(a,b)∂a-b∂DSH-SA(a,b)∂b)=(a-b)(lna-lnb)(a+b2(a2+b2)-12-2ab(a+b)2).
From (2.2) we have Λ≥0, so by Lemma 2.4, it follows that DSH-SA is Schur-geometrically convex in R+2.
(4) For(3.9)DAG-SG(a,b)=MAG(a,b)-12MSG(a,b)=12(a+b-ab-a2+b22),
we have
(3.10)∂DAG-SG(a,b)∂a=12(1-b2ab-a2(a2+b2)),∂DAG-SG(a,b)∂b=12(1-a2ab-b2(a2+b2)),
and then
(3.11)Λ∶=(lna-lnb)(a∂DSH-SA(a,b)∂a-b∂DSH-SA(a,b)∂b)=(a-b)(lna-lnb)(1-a+b2(a2+b2)).
By (2.2) we infer that
(3.12)1-a+b2(a2+b2)≥0,
so Λ≥0. By Lemma 2.4, we get that DAG-SG is Schur-geometrically convex in R+2.
(5) For(3.13)DN2N1-AH(a,b)=MN2N1(a,b)-18MAH(a,b)=(a+b2)(a+b2)-14(a+b)-12ab-18(a+b2-2aba+b),
we have
(3.14)∂DN2N1-AH(a,b)∂a=14aa+b2+14(a+b2)(a+b2)-1/2-14-b4ab-18(12-2b2(a+b)2),∂DN2N1-AH(a,b)∂b=14ba+b2+14(a+b2)(a+b2)-1/2-14-a4ab-18(12-2a2(a+b)2),
and then
(3.15)Λ=(lna-lnb)(a∂DN2N1-AH(a,b)∂a-b∂DN2N1-AH(a,b)∂b)=14(a-b)(lna-lnb)(a+b2(a+b)+a+b2a+b-54-ab(a+b)2).
From (2.4) we have
(3.16)a+b2(a+b)+a+b2a+b-54-ab(a+b)2≥0,
so Λ≥0; it follows that DN2N1-AH is Schur-geometrically convex in R+2.
(6) For(3.17)DN2G-N2N1(a,b)=13MN2G(a,b)-MN2N1(a,b)=a+b4+ab6-23(a+b2)(a+b2),
we have
(3.18)∂DN2G-N2N1(a,b)∂a=14+b12ab-16aa+b2-16(a+b2)(a+b2)-1/2,∂DN2G-N2N1(a,b)∂b=14+a12ab-16ba+b2-16(a+b2)(a+b2)-1/2,
and then
(3.19)Λ=(lna-lnb)(a∂DN2G-N2N1(a,b)∂a-b∂DN2G-N2N1(a,b)∂b)=(lna-lnb)(14(a-b)-a-b6a+b2-(a-b)(a+b)12(a+b2)-1/2)=16(a-b)(lna-lnb)(32-a+b2(a+b)-a+b2a+b).
By (2.4) we infer that Λ≥0, which proves that DN2G-N2N1 is Schur-geometrically convex in R+2.
(7) For(3.20)DAG-N2G(a,b)=14MAG(a,b)-13MN2G(a,b)=a+b8+112ab-13(a+b2)(a+b2),
we have
(3.21)∂DAG-N2G(a,b)∂a=18+b24ab-a+b122a-a+b122(a+b),∂DAG-N2G(a,b)∂b=18+a24ab-a+b122b-a+b122(a+b),
and then
(3.22)Λ=(lna-lnb)(a∂DAG-N2G(a,b)∂a-b∂DAG-N2G(a,b)∂b)=(lna-lnb)(a-b8-a+b(a-b)122-(a-b)(a+b)122(a+b))=(a-b)(lna-lnb)8(1-23(a+b2(a+b)+a+b2a+b)).
From (2.4) we have Λ≥0, and, consequently, by Lemma 2.4, we obtain that DAG-N2G is Schur-geometrically convex in R+2.
(8) In order to prove that the functionDAN2-AG(a,b) is Schur-geometrically convex in R+2 it is enough to notice that
(3.23)DAN2-AG(a,b)=MAN2(a,b)-14MAG(a,b)=3DAG-N2G(a,b).
(9) For(3.24)DSN2-SA(a,b)=45MSN2(a,b)-MSA(a,b)=a+b2-15a2+b22-15(a+b)2(a+b),
we have
(3.25)∂DSN2-SA(a,b)∂a=12-a52(a2+b2)-15a+b2a-a+b52(a+b),∂DSN2-SA(a,b)∂b=12-b52(a2+b2)-15a+b2b-a+b52(a+b),
and then
(3.26)Λ=(lna-lnb)(∂DSN2-SA(a,b)∂a-∂DSN2-SA(a,b)∂b)=(lna-lnb)(a-b2-a2-b252(a2+b2)-15(a(a+b)2-b(a+b)2)-(a+b)(a-b)52(a+b))=(a-b)(lna-lnb)52(52-a+ba2+b2-a+ba+b-a+ba+b).
From (2.2) and (2.4) we obtain that
(3.27)52-a+ba2+b2-a+ba+b-a+ba+b≥52-2-32=0,
so Λ≥0, which proves that the function DSN2-SA(a,b) is Schur-geometrically convex in R+2.
(10) One can easily check that(3.28)DANAN2-SN2(a,b)=4DSN2-SA(a,b),
and, consequently, the function DAN2-SN2 is Schur-geometrically convex in R+2.
(11) To prove that the function(3.29)DSN1-SH(a,b)=2MSN1(a,b)-MSH(a,b)=a2+b22-a+b2-ab+2aba+b
is Schur-geometrically convex in R+2 it is enough to notice that
(3.30)DSN1-SH(a,b)=DSG-AH(a,b).
(12) For(3.31)DSG-SN1(a,b)=32MSG(a,b)-2MSN1(a,b)=12(a+b-ab-a2+b22),
we have
(3.32)∂DSG-SN1(a,b)∂a=12(1-b2ab-a2(a2+b2)),∂DSG-SN1(a,b)∂b=12(1-a2ab-b2(a2+b2)),
and then
(3.33)Λ=(lna-lnb)(a∂DSG-SN1(a,b)∂a-b∂DSG-SN1(a,b)∂b)=(a-b)(lna-lnb)2(1-a+b2(a2+b2)).
By the inequality (2.2) we get that Λ≥0, which proves that DSG-SN1 is Schur-geometrically convex in R+2.
(13) It is easy to check that(3.34)DSN3-SA(a,b)=12DAG-SG(a,b),
which means that the function DSN3-SA is Schur-geometrically convex in R+2.
(14) To prove that the functionDSN1-SN3 is Schur-geometrically convex in R+2 it is enough to notice that
(3.35)DSN1-SN3(a,b)=16DAG-SG(a,b).
The proof of Theorem I is complete.


## 4. Applications

Applying Theorem I, Lemma2.6, and Definition 2.2 one can easily prove the following.Theorem II.
Let0<a≤b. 1/2≤t≤1 or 0≤t≤1/2, u=atb1-t and v=bta1-t. Then
(4.1)MSA(a,b)≤13MSH(a,b)-(13MSH(u,v)-MSA(u,v))≤13MSH(a,b)≤12MAH(a,b)-(12MAH(u,v)-13MSH(u,v))≤12MAH(a,b)≤12MSG(a,b)-(12MSG(u,v)-12MAH(u,v))≤12MSG(a,b)≤MAG(a,b)-(MAG(u,v)-12MSG(u,v))≤MAG(a,b),(4.2)18MAH(a,b)≤MN2N1(a,b)-(MN2N1(u,v)-18MAH(u,v))≤MN2N1(a,b)≤13MN2G(a,b)-(13MN2G(u,v)-MN2N1(u,v))≤13MN2G(a,b)≤14MAG(a,b)-(14MAG(u,v)-13MN2G(u,v))≤14MAG(a,b)≤MAN2(a,b)-(MAN2(u,v)-14MAG(u,v))≤MAN2(a,b),(4.3)MSA(a,b)≤45MSN2(a,b)-(45MSN2(u,v)-45MSN2(u,v))≤45MSN2(a,b)≤4MAN2(a,b)-(4MAN2(u,v)-45MSN2(u,v))≤4MAN2(a,b),(4.4)MSH(a,b)≤2MSN1(a,b)-(2MSN1(u,v)-MSH(u,v))≤2MSN1(a,b)≤32MSG(a,b)-(32MSG(u,v)-32MSG(u,v))≤32MSG(a,b),(4.5)MSA(a,b)≤34MSN3(a,b)-(34MSN3(u,v)-MSA(u,v))≤34MSN3(a,b)≤23MSN1(a,b)-(23MSN1(u,v)-34MSN3(u,v))≤23MSN1(a,b).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.



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