# Sharp Generalized Seiffert Mean Bounds for Toader Mean

Authors: Yu-Ming Chu; Miao-Kun Wang; Song-Liang Qiu; Ye-Fang Qiu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Forp∈[0,1], the generalized Seiffert mean of two positive numbers a and b is defined by Sp(a,b)=p(a-b)/arctan[2p(a-b)/(a+b)],0<p≤1,a≠b;(a+b)/2,p=0,a≠b;a,a=b. In this paper, we find the greatest value α and least value β such that the double inequality Sα(a,b)<T(a,b)<Sβ(a,b) holds for all a,b>0 with a≠b, and give new bounds for the complete elliptic integrals of the second kind. Here, T(a,b)=(2/π)∫0π/2a2cos2θ+b2sin2θdθ denotes the Toader mean of two positive numbers a and b.


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## Body

## 1. Introduction

Forp∈[0,1],  the generalized Seiffert mean of two positive numbers a and b is defined by
(1.1)Sp(a,b)={p(a-b)arctan[2p(a-b)/(a+b)],0<p≤1,a≠b,a+b2,p=0,a≠b,a,a=b.It is well known thatSp(a,b) is continuous and strictly increasing with respect to p∈[0,1] for fixed a,b>0 with a≠b. In particular, if p=1/2, then the generalized Seiffert mean reduces to the Seiffert mean
(1.2)S(a,b)={a-b2arctan((a-b)/(a+b)),a≠b,a,a=b.Recently, the Seiffert mean and its generalization have been the subject of intensive research, many remarkable inequalities for these means can be found in the literature [1–5].In [6], Toader introduced the Toader mean T(a,b) of two positive numbers a and b as follows:
(1.3)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 integral of the second kind.Vuorinen [7] conjectured that
(1.4)M3/2(a,b)<T(a,b)
for all a,b>0 with a≠b, where
(1.5)Mp(a,b)={(ap+bp2)1/p,p≠0,ab,p=0
is the power mean of order p of two positive numbers a and b. This conjecture was proved by Barnard et al. [8].In [9], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows:(1.6)T(a,b)<Mlog⁡2/log⁡(π/2)(a,b)
for all a,b>0 with a≠b.The main purpose of this paper is to find the greatest valueα and least value β such that the double inequality Sα(a,b)<T(a,b)<Sβ(a,b) holds for all a,b>0 with a≠b and give new bounds for the complete elliptic integrals of the second kind.


## 2. Lemmas

In order to establish our main result, we need several formulas and lemmas, which we present in this section.The following formulas were presented in [10, Appendix E, pages 474-475]: Letr∈[0,1), then
(2.1)K(r)=∫0π/2(1-r2sin⁡2t)-1/2dt,K(0)=π2,K(1-)=+∞,E(r)=∫0π/2(1-r2sin⁡2t)1/2dt,E(0)=π/2,E(1-)=1,dK(r)dr=E(r)-(1-r2)K(r)r(1-r2),dE(r)dr=E(r)-K(r)r,d[E(r)-(1-r2)K(r)]dr=rK(r),d[K(r)-E(r)]dr=rE(r)1-r2,E(2r1+r)=2E(r)-(1-r2)K(r)1+r.Lemma 2.1 (see [10, Theorem 1.25]).
For-∞<a<b<∞, let f(x),g(x):[a,b]→ℝ be continuous on [a,b] and be differentiable on (a,b), let g′(x)≠0 for all x∈(a,b). If f′(x)/g′(x) is increasing (decreasing) on (a,b), then so are
(2.2)f(x)-f(a)g(x)-g(a),f(x)-f(b)g(x)-g(b).
If f′(x)/g′(x) is strictly monotone, then the monotonicity in the conclusion is also strict.Lemma 2.2.
(1)[ℰ(r)-(1-r2)𝒦(r)]/r2 is strictly increasing from (0,1) onto (π/4,1);
(2){[ℰ(r)-(1-r2)𝒦(r)]/r2-π/4}/r2 is strictly increasing from (0,1) onto (π/32,1-π/4);
(3)[𝒦(r)-ℰ(r)]/r2 is strictly increasing from (0,1) onto (π/4,+∞);
(4)2ℰ(r)-(1-r2)𝒦(r) is strictly increasing from (0,1) onto (π/2,2);
(5)F(r)=[(2-r2)𝒦(r)-2ℰ(r)]/r4 is strictly increasing from (0,1) onto (π/16,+∞);
(6)G(r)=[4π-πr2-8ℰ(r)]/r4 is strictly increasing from (0,1) onto (3π/16,3π-8).Proof.
Parts (1)–(4) can be found in [10, Theorem3.21(1), Theorem  3.31(6), and Exercise 3.43(11) and (13)].
For part (5), clearlyF(1-)=+∞. Let F1(r)=(2-r2)𝒦(r)-2ℰ(r) and F2(r)=r4, then F(r)=F1(r)/F2(r), F1(0)=F2(0)=0 and
(2.3)F1′(r)F2′(r)=E(r)-(1-r2)K4r2(1-r2).
It follows from (2.3) and part (1) together with Lemma 2.1 that F(r) is strictly increasing in (0,1) and F(0+)=π/16.
For part (6), clearlyG(1-)=3π-8. Let G1(r)=4π-πr2-8ℰ(r) and G2(r)=r4, then G(r)=G1(r)/G2(r), G1(0)=G2(0)=0, and
(2.4)G1′(r)2G2′(r)=(2-r2)K(r)-2E(r)r4+[E(r)-(1-r2)K(r)]/r2-π/4r2.
From (2.4), parts (2) and (5) together with Lemma 2.1, we know that G(r) is strictly increasing in (0,1), and f(0+)=3π/16.Lemma 2.3.
(1)g(r)=arctan(3r/2)-3πr/{4[2ℰ(r)-(1-r2)𝒦(r)]} is strictly increasing from (0,1) onto (0,arctan(3/2)-3π/8).
(2)f(r)=arctanr-πr/{2[2ℰ(r)-(1-r2)𝒦(r)]}<0 for r∈(0,1).Proof.
For part (1), clearlyg(0+)=0 and g(1-)=arctan(3/2)-3π/8=0.0335⋯>0. Simple computation leads to
(2.5)g′(r)=234+3r2-3πE(r)4[2E(r)-(1-r2)K(r)]2=3r4E(r)4(4+3r2)[2E(r)-(1-r2)K(r)]2g1(r),
where g1(r)={8[2ℰ(r)-(1-r2)𝒦(r)]2-π(4+3r2)ℰ(r)}/[r4ℰ(r)].
Making use of Lemma2.2 (1), (2), and (6), we get
(2.6)g1(r)=8E(r)⋅[E(r)-(1-r2)K(r)r2]2+16{[E(r)-(1-r2)K(r)]/r2-π/4}r2-4π-πr2-8E(r)r4>16π⋅(π4)2+16⋅π32-(3π-8)=8-3π2>0.
Therefore, part (1) follows from (2.5) and (2.6) together with the limiting values of g(r) at r=0 and r=1.
For part (2), simple computations yield that(2.7)lim⁡r→0+f(r)=lim⁡r→1-f(r)=0,(2.8)f′(r)=f1(r)2(1+r2)[2E(r)-(1-r2)K(r)]2,
where f1(r)=2[2ℰ(r)-(1-r2)𝒦(r)]2-π(1+r2)ℰ(r). Note that
(2.9)lim⁡r→0+⁡f1(r)=0,(2.10)lim⁡r→1-f1(r)=8-2π>0,(2.11)f1′(r)=4[2E(r)-(1-r2)K(r)][E(r)-(1-r2)K(r)]r-2πrE(r)-π(1+r2)E(r)-K(r)r=rf2(r),
where f2(r)=4[2ℰ(r)-(1-r2)𝒦(r)][ℰ(r)-(1-r2)𝒦(r)]/r2-2πℰ(r)+π(1+r2)[𝒦(r)-ℰ(r)]/r2.
From Lemma2.2(1), (3), and (4) together with the monotonicity of ℰ(r) we know that f2(r) is strictly increasing in (0,1). Moreover,
(2.12)lim⁡r→0+f2(r)=-π24,(2.13)lim⁡r→1-f2(r)=+∞.
Equations (2.11)–(2.13) and the monotonicity of f2(r) lead to the conclusion that there exists r0∈(0,1) such that f1(r) is strictly decreasing in (0,r0) and strictly increasing in (r0,1).
It follows from (2.8)–(2.10) and the piecewise monotonicity of f1(r) that there exists r1∈(0,1) such that f(r) is strictly decreasing in (0,r1) and strictly increasing in (r1,1).
Therefore, part(2) follows from (2.7) and the piecewise monotonicity of f(r).


## 3. Main Result

Theorem 3.1.
InequalityS3/4(a,b)<T(a,b)<S1/2(a,b) holds for all a,b>0 with a≠b, and S3/4(a,b) and S1/2(a,b) are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean T(a,b), respectively.Proof.
Firstly, we prove that(3.1)S3/4(a,b)<T(a,b)<S1/2(a,b)
for all a,b>0 with a≠b.
Without loss of generality, we assume thata>b. Let t=b/a<1, r=(1-t)/(1+t). Then (1.1) and (1.3) lead to
(3.2)T(a,b)-S3/4(a,b)=2aπE(1-t2)-3a(1-t)4arctan[3(1-t)/2(1+t)]=2aπE(2r1+r)-3ar2(1+r)arctan((3/2)r)=2aπ[2E(r)-(1-r2)K(r)]1+r-3ar2(1+r)arctan((3/2)r)=2a[2E(r)-(1-r2)K(r)]π(1+r)arctan((3/2)r)g(r),(3.3)T(a,b)-S1/2(a,b)=2aπE(1-t2)-a(1-t)2arctan((1-t)/(1+t))=2aπE(2r1+r)-ar(1+r)arctanr=2aπ[2E(r)-(1-r2)K(r)]1+r-ar(1+r)arctanr=2a[2E(r)-(1-r2)K(r)]π(1+r)arctanrf(r),
where g(r) and f(r) are defined as in Lemma 2.3.
Therefore, inequality (3.1) follows from (3.2) and (3.3) together with Lemma 2.3.
Next, we prove thatS3/4(a,b) and S1/2(a,b) are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean T(a,b), respectively.
For anyɛ>0 and 0<x<1, from (1.1) and (1.3) one has
(3.4)lim⁡x→0[S1/2-ɛ(1,x)-T(1,x)]=1-2ɛ2arctan(1-2ɛ)-2π<12arctan1-2π=0,(3.5)S3/4+ɛ(1,1-x)-T(1,1-x)=J(x)arctan[(3+4ɛ)x/2(2-x)],
where J(x)=(3/4+ɛ)x-2ℰ(2x-x2)arctan{[(3+4ɛ)x]/[2(2-x)]}/π.
Lettingx→0 and making use of Taylor expansion, we get
(3.6)J(x)=(34+ɛ)x-(34+ɛ)x[1-12x+116x2+o(x2)]×{1+12x+[14-13(34+ɛ)2]x2+o(x2)}=ɛ3(32+ɛ)(34+ɛ)x3+o(x3).
Inequality (3.4) and equations (3.5) and (3.6) imply that for any ɛ>0 there exist δ1=δ1(ɛ)>0 and δ2=δ2(ɛ)>0, such that S3/4+ɛ(1,1-x)>T(1,1-x) for x∈(0,δ1) and S1/2-ɛ(1,x)<T(1,x) for x∈(0,δ2).From Theorem3.1, we get new bounds for the complete elliptic integrals of the second kind as follows.Corollary 3.2.
The inequality(3.7)3π(1-1-r2)8arctan{3(1-1-r2)/[2(1+1-r2)]}<E(r)<π(1-1-r2)4arctan[(1-1-r2)/(1+1-r2)]
holds for all r∈(0,1).



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*Source: 605259-2011-10-13.xml*