# Inequalities between Arithmetic-Geometric, Gini, and Toader Means

Authors: Yu-Ming Chu; Miao-Kun Wang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We find the greatest valuesp1, p2 and least values q1, q2 such that the double inequalities Sp1(a,b)<M(a,b)<Sq1(a,b) and Sp2(a,b)<T(a,b)<Sq2(a,b) hold for all a,b>0 with a≠b and present some new bounds for the complete elliptic integrals. Here M(a,b), T(a,b), and Sp(a,b) are the arithmetic-geometric, Toader, and pth Gini means of two positive numbers a and b, respectively.


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## Body

## 1. Introduction

Forp∈ℝ the pth Gini mean Sp(a,b) and power mean Mp(a,b) of two positive real numbers a and b are defined by(1.1)Sp(a,b)={(ap-1+bp-1a+b)1/(p-2),p≠2,(aabb)1/(a+b),p=2,(1.2)Mp(a,b)={(ap+bp2)1/p,p≠0,ab,p=0,
respectively.It is well known thatSp(a,b) and Mp(a,b) are continuous and strictly increasing with respect to p∈ℝ for fixed a,b>0 with a≠b. Many means are special case of these means, for example,(1.3)S1(a,b)=M1(a,b)=a+b2=A(a,b)isthearithmeticmean,S0(a,b)=M0(a,b)=ab=G(a,b)isthegeometricmean,M-1(a,b)=2aba+b=H(a,b)istheharmonicmean.Recently, the Gini and power means have been the subject of intensive research. In particular, many remarkable inequalities for these means can be found in the literature [1–7].In [8], Toader introduced the Toader mean T(a,b) of two positive numbers a and b as follows:(1.4)T(a,b)=2π∫0π/2a2cos⁡2θ+b2sin⁡2θdθ={2aE(1-(b/a)2)π,a>b,2bE(1-(a/b)2)π,a<b,a,a=b,
where ℰ(r)=∫0π/2(1-r2sin2t)1/2dt, r∈[0,1], is the complete elliptic integrals of the second kind.The classical arithmetic-geometric meanM(a,b) of two positive number a and b is defined as the common limit of sequences {an} and {bn}, which are given by(1.5)a0=a,b0=b,an+1=an+bn2=A(an,bn),bn+1=anbn=G(an,bn).The Gauss identity [9] shows that(1.6)M(1,r)K(1-r2)=π2
for r∈(0,1), where 𝒦(r)=∫0π/2(1-r2sin2t)-1/2dt, r∈[0,1), is the complete elliptic integrals of the first kind.Vuorinen [10] conjectured that(1.7)M3/2(a,b)<T(a,b)
for all a,b>0 with a≠b. This conjecture was proved by Qiu and Shen in [11] and Barnard et al. in [12], respectively.In [13], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows:(1.8)T(a,b)<Mlog⁡2/log⁡(π/2)(a,b)
for all a,b>0 with a≠b.In [14–17], the authors proved that(1.9)M0(a,b)=G(a,b)<M(a,b)<M1/2(a,b),(1.10)L(a,b)<M(a,b)<π2L(a,b)
for all a,b>0 with a≠b, where(1.11)L(a,b)={a-blog⁡a-log⁡b,a≠b,a,a=b,
denotes the classical logarithmic mean of two positive numbers a and b.Very recently, Chu and Wang [18] and Guo and Qi [19] proved that(1.12)L0(a,b)<T(a,b)<L1/4(a,b)
for all a,b>0 with a≠b, and L0(a,b) and L1/4(a,b) are the best possible lower and upper Lehmer mean bounds for the Toader mean T(a,b), respectively. Here, the pth Lehmer mean Lp(a,b) of two positive numbers a and b is defined by Lp(a,b)=(ap+1+bp+1)/(ap+bp).The main purpose of this paper is to find the greatest valuesp1, p2 and least values q1, q2 such that the double inequalities Sp1(a,b)<M(a,b)<Sq1(a,b) and Sp2(a,b)<T(a,b)<Sq2(a,b) hold for all a,b>0 with a≠b and present some new bounds for the complete elliptic integrals.


## 2. Preliminary Knowledge

Throughout this paper, we denoter′=1-r2 for r∈[0,1].For0<r<1, the following derivative formulas were presented in [9, Appendix E, pages 474–475]:(2.1)dK(r)dr=E(r)-r′2K(r)rr′2,dE(r)dr=E(r)-K(r)r,d[E(r)-r′2K(r)]dr=rK(r),d[K(r)-E(r)]dr=rE(r)r′2.(2.2)K(2r1+r)=(1+r)K(r),(2.3)E(2r1+r)=2E(r)-r′2K(r)1+r.Lemma2.1 can be found in [9, Theorem  3.21(7), (8), and (10), and Exercise  3.43(13) and (46)].Lemma 2.1.
(1)r′c𝒦(r) is strictly decreasing from [0,1) onto (0,π/2] for c∈[1/2,∞);
(2)r′cℰ(r) is strictly increasing on (0,1) if and only if c≤-1/2 and strictly decreasing if and only if c>0;
(3)𝒦(r)/log(4/r′) is strictly decreasing from (0,1) onto (1,π/log16);
(4)2ℰ(r)-r′2𝒦(r) is strictly increasing from (0,1) onto (π/2,2);
(5)[ℰ(r)-r′2𝒦(r)]/[r2𝒦(r)] is strictly decreasing from (0,1) onto (0,1/2).


## 3. Main Results

Theorem 3.1.
InequalityS1/2(a,b)<M(a,b)<S1(a,b) holds for all a,b>0 with a≠b, and S1/2(a,b) and S1(a,b) are the best possible lower and upper Gini mean bounds for the arithmetic-geometric mean M(a,b).Proof.
From (1.1) and (1.5) we clearly see that both Sp(a,b) and M(a,b) are symmetric and homogenous of degree 1. Without loss of generality, we assume that a=1>b. Let t=b and r=(1-t)/(1+t). Then from (1.1) and (1.6) together with (2.2) we clearly see that
(3.1)M(a,b)-S1/2(a,b)=π2K(1-t2)-[(1+t)t1+t]2/3=π2(1+r)K(r)-[21-r(1+r)(1+r+1-r)]2/3=11+r[π2K(r)-(2r′1+r+1-r)2/3].
Let(3.2)F(r)=[π2K(r)]3-(2r′1+r+1-r)2.
Then F(r) can be rewritten as
(3.3)F(r)=[π2K(r)]3-2r′21+r′=2r′21+r′F1(r),
where
(3.4)F1(r)=(π2)31+r′2r′2K(r)3-1.
It is well known that the functionr→r+1/r is positive and strictly decreasing in (0,1). Then (3.4) and Lemma 2.1(1) lead to the conclusion that F1(r) is strictly increasing in (0,1), so that F1(r)>F1(0)=0 for r∈(0,1).
Therefore,M(a,b)>S1/2(a,b) follows from (3.1)–(3.3).
On the other hand,M(a,b)<S1(a,b)=A(a,b) follows directly from (1.9).
Next, we prove thatS1/2(a,b) and S1(a,b) are the best possible lower and upper Gini mean bounds for the arithmetic-geometric mean M(a,b).
For any0<ε<1/2 and 0<x<1, from (1.1), (1.6), and Lemma 2.1(3) we have
(3.5)[M(1,1-x)]3-2ε-[S1/2+ε(1,1-x)]3-2ε=[π2∫0π/2[1-(2x-x2)sin⁡2t]-1/2dt]3-2ε-[(2-x)(1-x)1/2-ε1+(1-x)1/2-ε]2,(3.6)lim⁡x→0M(1,x)S1-ε(1,x)=lim⁡x→0[2πxε/(1+ε)K(1-x2)(1+xε1+x)1/(1+ε)]=lim⁡x→02πxε/(1+ε)K(1-x2)=lim⁡x→02πxε/(1+ε)log⁡(4/x)log⁡(4/x)K(1-x2)=lim⁡x→02πxε/(1+ε)log⁡(4/x)=+∞.
Letting x→0 and making use of the Taylor expansion, one has
(3.7)[π2∫0π/2[1-(2x-x2)sin⁡2t]-1/2dt]3-2ε-[(2-x)(1-x)1/2-ε1+(1-x)1/2-ε]2=1+(-32+ε)x+(2ε-3)(4ε-3)16x2+o(x2)-[1+(-32+ε)x+(2ε-3)216x2+o(x2)]=-ε(3-2ε)8x2+o(x2).
Equations (3.5)–(3.7) imply that for any 1<ε<1/2 there exist δ1=δ1(ε)∈(0,1) and δ2=δ2(ε)∈(0,1), such that M(1,1-x)<S1/2+ε(1,1-x) for x∈(0,δ1) and M(1,x)>S1-ε(1,x) for x∈(0,δ2).Theorem 3.2.
InequalityS1(a,b)<T(a,b)<S3/2(a,b) holds for all a,b>0 with a≠b, and S1(a,b) and S3/2(a,b) are the best possible lower and upper Gini mean bounds for the Toader mean T(a,b).Proof.
From (1.1) and (1.4) we clearly see that both Sp(a,b) and T(a,b) are symmetric and homogenous of degree 1. Without loss of generality, we assume that a=1>b. Let t=b and r=(1-t)/(1+t). Then from (1.1), (1.4), and (2.3) we have
(3.8)T(a,b)S3/2(a,b)=2πE(1-t2)⋅(1+t1+t)2=2πE(2r1+r)⋅(1+r)⋅(1+r+1-r2)2=2π[2E(r)-r′2K(r)]⋅(1+r+1-r2)2=1π(1+r′)[2E(r)-r′2K(r)].
Let(3.9)G(r)=1π(1+r′)[2E(r)-r′2K(r)].
Then simple computations lead to
(3.10)G(0)=1,(3.11)G′(r)=1π[(-rr′)(2E(r)-r′2K(r))+(1+r′)(E(r)-r′2K(r)r)]=r′(1+r′)[E(r)-r′2K(r)]-r2[2E(r)-r′2K(r)]πrr′=rπr′G1(r),
where
(3.12)G1(r)=(1+r′)r′K(r)[E(r)-r′2K(r)r2K(r)]-[2E(r)-r′2K(r)].
It follows from (3.12) and Lemma 2.1(1), (4), and (5) that G1(r) is strictly decreasing from (0,1) onto (-2,0). Then (3.11) leads to the conclusion that G′(r)<0 for r∈(0,1). Hence G(r) is strictly decreasing in (0,1).
Therefore,T(a,b)<S3/2(a,b) follows from (3.8)–(3.10) together with the monotonicity of G(r).
On the other hand,T(a,b)>S1(a,b)=A(a,b) follows directly from (1.7).
Next, we prove thatS1(a,b) and S3/2(a,b) are the best possible lower and upper Gini mean bounds for the Toader mean T(a,b).
For any0<ε<1/2 and 0<x<1, from (1.1) and (1.4) one has
(3.13)[T(1,1-x)]1+2ε-[S3/2-ε(1,1-x)]1+2ε=[2π∫0π/2[1-(2x-x2)sin⁡2t]1/2dt]1+2ε-[2-x1+(1-x)1/2-ε]2,(3.14)lim⁡x→0T(1,x)S1+ε(1,x)=lim⁡x→0[2πE(1-x2)(1+xε1+x)1/(1-ε)]=2π<1.
Lettingx→0 and making use of the Taylor expansion, we get
(3.15)[2π∫0π/2[1-(2x-x2)sin⁡2t]1/2dt]1+2ε-[2-x1+(1-x)1/2-ε]2=1-(12+ε)x+(2ε+1)(4ε+1)16x2+o(x2)-[1-(12+ε)x+(2ε+1)216x2+o(x2)]=ε(2ε+1)8x2+o(x2).
Equations (3.13)–(3.15) imply that for any 0<ε<1/2  there exist δ3=δ3(ε)∈(0,1) and δ4=δ4(ε)∈(0,1), such that  T(1,1-x)>S3/2-ε(1,1-x) for x∈(0,δ3) and T(1,x)<S1+ε(1,x) for x∈(0,δ4).


## 4. Remarks and Corollaries

Remark 4.1.
From (3.9) and Lemma 2.1(4) we clearly see that G(1-)=2/π. Then (3.8) and (3.9) together with the monotonicity of G(r) lead to the conclusion that
(4.1)2πS3/2(a,b)<T(a,b)
for all a,b>0 with a≠b.Remark 4.2.
We find that the lower boundL(a,b) in (1.10) and the best possible lower Gini mean bound S1/2(a,b) in Theorem 3.1 are not comparable. In fact, from (1.1) and (1.11) we have
(4.2)lim⁡x→+∞S1/2(1,x)L(1,x)=lim⁡x→+∞[1+x-11+x-1/2]2/3x2/3log⁡xx-1=lim⁡x→+∞log⁡xx1/3-x-2/3=0,S1/2(1,1+x)-L(1,1+x)=1+12x-116x2+o(x2)-[1+12x-112x2+o(x2)]=148x2+o(x2)(x⟶0).Remark 4.3.
The following two equations show that the best possible upper power mean boundMlog2/log(π/2)(a,b) in (1.8) and the best possible upper Gini mean bound S3/2(a,b) in Theorem 3.2 are not comparable:
(4.3)lim⁡x→+∞S3/2(1,x)Mlog⁡2/log⁡(π/2)(1,x)=2log⁡(π/2)/log⁡2=π2,Mlog⁡2/log⁡(π/2)(1,1+x)-S3/2(1,1+x)=1+12x+18[log⁡2log⁡(π/2)-1]x2+o(x2)-[1+12x+116x2+o(x2)]=116[2log⁡2log⁡(π/2)-3]x2+o(x2)=0.00436⋯×x2+o(x2)(x⟶0).From Theorem3.1 we get an upper bound for the complete elliptic integrals of the first kind 𝒦(r) as follows.Corollary 4.4.
Inequality(4.4)K(r)<π2[1+(1-r2)1/4(1+1-r2)(1-r2)1/4]2/3
holds for all r∈(0,1).Remark 4.5.
Computational and numerical experiments show that the upper bound in (4.4) for 𝒦(r) is very accurate for some r∈(0,1). In fact, if we let H(r)=π[1+(1-r2)1/4]2/3/{2[(1+1-r2)(1-r2)1/4]2/3}, then we have Table 1 via elementary computation.Table 1
Comparison of𝒦(r) with H(r) for some r∈(0,1).
r𝒦(r)H(r)0.11.574745561517⋯1.574745561518⋯0.21.586867847⋯1.586867848⋯0.31.608048620⋯1.608048634⋯0.41.639999866⋯1.640000021⋯0.51.685750355⋯1.685751528⋯0.61.750753803⋯1.750760840⋯0.71.845693998⋯1.845732233⋯0.81.995302778⋯1.995519211⋯The following bounds for the complete elliptic integrals of the second kindℰ(r) follow from Theorem 3.2 and Remark 4.1.Corollary 4.6.
Inequality(4.5)[1+1-r21+(1-r2)1/4]2<E(r)<π2[1+1-r21+(1-r2)1/4]2
holds for all r∈(0,1).Remark 4.7.
Computational and numerical experiments show that the upper bound in (4.5) for ℰ(r) is very accurate for some r∈(0,1). In fact, if we let J(r)=π[1+1-r2]2/{2[1+(1-r2)1/4]2}, then we have Table 2 via elementary computation.Table 2
Comparison ofℰ(r) with J(r) for some r∈(0,1).
rℰ(r)J(r)0.11.566861942021⋯1.566861942028⋯0.21.554968546⋯1.554968548⋯0.31.534833465⋯1.534833516⋯0.41.505941612⋯1.505942206⋯0.51.467462209⋯1.467466484⋯0.61.418083394⋯1.418107161⋯0.71.355661136⋯1.355777213⋯0.81.276349943⋯1.276910677⋯



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