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Journal of Inequalities and Applications
 
J Inequal Appl. 2017; 2017(1): 201.
Published online 2017 August 29. doi:  10.1186/s13660-017-1473-1
PMCID: PMC5575000

Optimal convex combination bounds of geometric and Neuman means for Toader-type mean

Abstract

In this paper, we prove that the double inequalities

αNQA(a,b)+(1α)G(a,b)<TD[A(a,b),G(a,b)]<βNQA(a,b)+(1β)G(a,b),λNAQ(a,b)+(1λ)G(a,b)<TD[A(a,b),G(a,b)]<μNAQ(a,b)+(1μ)G(a,b)

hold for all ab > 0 with a ≠ b if and only if α ≤ 3/8, β4/[π(log(1+2)+2)]=0.5546 , λ ≤ 3/10 and μ ≥ 8/[π(π + 2)] = 0.4952 ⋯  , where TD(ab), G(ab), A(ab) and NQA(ab), NAQ(ab) are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively.

Keywords: Toader mean, geometric mean, Neuman mean

Introduction

For xyz ≥ 0 with xyxzyz ≠ 0 and r ∈ (0, 1), the symmetric integrals RF(xyz) and RG(xyz) [1] of the first and second kinds, and the complete elliptic integrals 𝒦(r) and ℰ(r) of the first and second kinds are defined by

RF(x,y,z)=120[(t+x)(t+y)(t+z)]1/2dt,RG(x,y,z)=140[(t+x)(t+y)(t+z)]1/2(xt+x+yt+y+zt+z)tdt,K(r)=0π/2[1r2sin2(t)]1/2dt,E(r)=0π/2[1r2sin2(t)]1/2dt,

respectively.

The well-known identities

𝒦(r) = RF(0, 1 − r2, 1),  ℰ(r) = 2RG(0, 1 − r2, 1)

were established by Carlson in [1].

Let ab > 0 with a ≠ b. Then the Toader mean TD(ab) [2] and the Schwab-Borchardt mean SB(ab) [35] are respectively defined by

TD(a,b)=2π0π/2a2cos2(t)+b2sin2(t)dt={2aE(1(b/a)2)/π,a>b,2bE(1(a/b)2)/π,a<b,
1.1

and

SB(a,b)={b2a2cos1(a/b),a<b,a2b2cosh1(a/b),a>b,

where cos−1(x) and cosh1(x)=log(x+x21) are the inverse cosine and inverse hyperbolic cosine functions, respectively.

Very recently, Neuman [6] introduced the Neuman mean N(ab) of the second kind as follows:

N(a,b)=12[a+b2SB(a,b)].

It is well known that the Toader mean TD(ab), the Schwab-Borchardt mean SB(ab) and the Neuman mean of the second kind N(ab) satisfy the identities (see [6, 7])

TD(a,b)=4πRG(a2,b2,0)=1π0[(t+a2)(t+b2)]1/2(a2t+a2+b2t+b2)tdt,SB(a,b)=1/RF(a2,b2,b2)=2/0[(t+a2)(t+b2)(t+b2)]1/2dt,N(a,b)=RG(a2,b2,b2)=140[(t+a2)(t+b2)(t+b2)]1/2(a2t+a2+b2t+b2+b2t+b2)tdt.

Let p ∈ ℝ and ab > 0. Then the pth power mean Mp(ab) is defined by

Mp(a,b)=[(ap+bp)/2]1/p(p0),M0(a,b)=ab.
1.2

We clearly see that Mp(ab) is symmetric and homogeneous of degree one with respect to a and b, strictly increasing with respect to p ∈ ℝ for fixed ab > 0 with a ≠ b, and the inequalities

G(ab) = M0(ab) < A(ab) = M1(ab) < Q(ab) = M2(ab)

hold for ab > 0 with a ≠ b, where G(a,b)=ab, A(ab) = (ab)/2 and Q(a,b)=(a2+b2)/2 are the geometric, arithmetic and quadratic means of a and b, respectively.

In [6], Neuman presented the explicit formula for NQA(ab) ≡ N[Q(ab), A(ab)] and NAQ(ab) ≡ N[A(ab), Q(ab)] as follows:

NQA(a,b)=12A(a,b)[1+v2+sinh1(v)v],
1.3

NAQ(a,b)=12A(a,b)[1+(1+v2)tan1(v)v]
1.4

and proved that the inequalities

A(ab) < NQA(ab) < NAQ(ab) < Q(ab)
1.5

hold for ab > 0 with a ≠ b, where v = (a − b)/(ab).

Recently, the Toader mean has been the subject of intensive research. In particular, many remarkable inequalities for Toader mean and other related means can be found in the literature [841].

In [42], Vuorinen conjectured that

TD(ab) > M3/2(ab)

for all ab > 0 with a ≠ b. This conjecture was proved by Qiu and Shen [43], and Barnard et al. [44], respectively, and Alzer and Qiu [45] presented the best possible upper power mean bound for the Toader mean as follows:

TD(ab) < Mlog2/log(π/2)(ab)

for all ab > 0 with a ≠ b.

Li, Qian and Chu [46] proved that the inequality

αNAQ(ab) + (1 − α)A(ab) < TD(ab) < βNAQ(ab) + (1 − β)A(ab)

holds for all ab > 0 with a ≠ b if and only if α ≤ 3/4 and β ≥ 4(4 − π)/[π(π − 2)] = 0.9573 ⋯  .

Note that

G(ab) < TD[A(ab), G(ab)] < A(ab)
1.6

for all ab > 0 with a ≠ b.

From inequalities (1.5) and (1.6) we clearly see that

G(ab) < TD[A(ab), G(ab)] < NQA(ab) < NAQ(ab)

for all ab > 0 with a ≠ b.

The main purpose of this paper is to find the greatest values α, λ and the least values β, μ such that the double inequalities

αNQA(a,b)+(1α)G(a,b)<TD[A(a,b),G(a,b)]<βNQA(a,b)+(1β)G(a,b),λNAQ(a,b)+(1λ)G(a,b)<TD[A(a,b),G(a,b)]<μNAQ(a,b)+(1μ)G(a,b)

hold for all ab > 0 with a ≠ b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions.

Lemmas

In order to prove our main results, we need several lemmas, which we present in this section.

For r ∈ (0, 1), we clearly see that

𝒦(0+) = ℰ(0+) = π/2,  𝒦(1) =  + ∞,  ℰ(1) = 1, 

and 𝒦(r) and ℰ(r) satisfy the formulas (see[21], Appendix E, pp.474-475)

dK(r)dr=E(r)(1r2)K(r)r(1r2),dE(r)dr=E(r)K(r)r,d[E(r)K(r)]dr=rE(r)1r2.

Lemma 2.1

see [21], Theorem 1.25

For −∞ < a < b <  + ∞, let fg:[ab] → ℝ be continuous on [ab] and differentiable on (ab), and g(x) ≠ 0 on (ab). If f(x)/g(x) is increasing (decreasing) on (ab), then so are

f(x)f(a)g(x)g(a)andf(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

see [21], Theorem 3.21(1), Exercise 3.43(11) and Exercise 3.43(29)

  1. The function r ↦ [ℰ(r)−(1 − r2)𝒦(r)]/r2 is strictly increasing from (0, 1) onto (π/4, 1);
  2. The function r ↦ [𝒦(r)−ℰ(r)]/r2 is strictly increasing from (0, 1) onto (π/4,  + ∞);
  3. The function r ↦ [(2 − r2)𝒦(r)−2ℰ(r)]/r4 is strictly increasing from (0, 1) onto (π/16,  + ∞).

Lemma 2.3

The function rφ1(r)={2π1r2[2E(r)K(r)]+2r21}/r2 is strictly increasing from (0, 1) onto (3/4, 1).

Proof

Simple computations lead to

φ1(0+) = ¾,  φ1(1) = 1, 
2.1

φ1(r)=2πr3γ1(r),
2.2

where

γ1(r)=K(r)3E(r)1r2+π,
2.3

γ1(0+) = 0, 
2.4

γ1(r)=r3(1r2)3/2(2r2)K(r)2E(r)r4.
2.5

From (2.5) and Lemma 2.2(3) we get

γ1(r)>πr316(1r2)3/2>0.
2.6

Therefore, Lemma 2.3 follows easily from (2.1), (2.2), (2.4) and (2.6).

Lemma 2.4

The function rφ2(r)=(2r2+1r41)/r2 is strictly decreasing from (0, 1) onto (1, 2).

Proof

It is easy to verify that

φ2(0+) = 2,  φ2(1) = 1, 
2.7

φ2(r)=2(1r41)r31r4<0
2.8

for r ∈ (0, 1).

Therefore, Lemma 2.4 follows easily from (2.7) and (2.8).

Lemma 2.5

The function rφ3(r)=[2r2K(r)5E(r)]/1r2 is strictly increasing from (0, 1) onto (−5π/2,  + ∞).

Proof

It is not difficult to verify that

φ3(0+)=52π,φ3(1)=+,
2.9

φ3(r)=r(1r2)3/2[(53r2)K(r)E(r)r2E(r)].
2.10

From (2.10) and Lemma 2.2(2) together with the monotonicity of ℰ(r) on (0, 1) we clearly see that

φ3(r)>r(1r2)3/2[(53r2)×π4π2]=3π4r1r2>0
2.11

for r ∈ (0, 1).

Therefore, Lemma 2.5 follows from (2.9) and (2.11).

Lemma 2.6

The function rφ4(r)={2π1r2[2E(r)(1+r2)K(r)]+3r21}/r2 is strictly increasing from (0, 1) onto (3/4, 2).

Proof

Let ϕ1(r)=2π1r2[2E(r)(1+r2)K(r)]+3r21, ϕ2(r) = r2. Then simple computations give

ϕ1(0+) = ϕ2(0) = 0,  φ4(r) = ϕ1(r)/ϕ2(r), 
2.12

φ4(1) = 2, 
2.13

ϕ1(r)ϕ2(r)=3+1π1r2[E(r)(1r2)K(r)r2]+1πφ3(r).
2.14

It follows from Lemma 2.2(1), Lemma 2.5 and the function r1r2 strictly decreasing that ϕ1(r)/ϕ2(r) is strictly increasing on (0, 1) and

φ4(0+)=limr0+ϕ1(r)ϕ2(r)=34.
2.15

Therefore, Lemma 2.6 follows from Lemma 2.1, (2.12), (2.13) and (2.15) together with the monotonicity of ϕ1(r)/ϕ2(r).

Lemma 2.7

The function φ5(r)=[3r2+1r21]/r2 is strictly decreasing from (0, 1) onto (2, 5/2).

Proof

We clearly see that

φ5(0+)=52,φ5(1)=2,
2.16

φ5(r)=(11r2)2r31r2<0
2.17

for r ∈ (0, 1).

Therefore, Lemma 2.7 follows easily from (2.16) and (2.17).

Main results

Theorem 3.1

The double inequality

αNQA(ab) + (1 − α)G(ab) < TD[A(ab), G(ab)] < βNQA(ab) + (1 − β)G(ab)
3.1

holds for all ab > 0 with a ≠ b if and only if α ≤ 3/8 and β4/[π(log(1+2)+2)]=0.5546 .

Proof

Since G(ab), TD(ab) and NQA(ab) are symmetric and homogenous of degree 1, without loss of generality, we assume that a > b > 0 and let r = (a − b)/(ab) ∈ (0, 1). Then (1.1)-(1.3) lead to

TD[A(a,b),G(a,b)]=2πA(a,b)E(r),
3.2

G(a,b)=A(a,b)1r2,NQA(a,b)=12A(a,b)[1+r2+sinh1(r)r].
3.3

It follows from (3.2)-(3.3) that

T[A(a,b),G(a,b)]G(a,b)NQA(a,b)G(a,b)=2πε(r)1r212[1+r2+sinh1(r)r]1r2=4πrε(r)2r1r2sinh1(r)+(r1+r22r1r2).
3.4

Let f1(r)=4πrε(r)2r1r2, f2(r)=sinh1(r)+(r1+r22r1r2) and

f(r)=4πrε(r)2r1r2sinh1(r)+(r1+r22r1r2).
3.5

Then simple computations lead to

f1(0+) = f2(0) = 0, 
3.6

f1(r)f2(r)=2π1r2[2ε(r)κ(r)]+2r212r2+1r41=φ1(r)φ2(r),
3.7

where φ1(r) and φ2(r) are defined as in Lemmas 2.3 and 2.4.

It follows from Lemmas 2.3-2.4 and (3.7) that f1(r)/f2(r) is strictly increasing on (0, 1). Then (3.5), (3.6) and Lemma 2.1 lead to the conclusion that f(r) is strictly increasing.

Moreover,

limr0+4πrε(r)2r1r2sinh1(r)+(r1+r22r1r2)=38,
3.8

limr14πrε(r)2r1r2sinh1(r)+(r1+r22r1r2)=4π[π(log(1+2)+2)].
3.9

Therefore, Theorem 3.1 follows easily from (3.4), (3.8) and (3.9) together with the monotonicity of f(r).

Theorem 3.2

The double inequality

λNAQ(a,b)+(1λ)G(a,b)<TD[A(a,b),G(a,b)]<μNAQ(a,b)+(1μ)G(a,b)
3.10

holds for all ab > 0 with a ≠ b if and only if λ ≤ 3/10 and μ ≥ 8/[π(π + 2)] = 0.4952 ⋯  .

Proof

Without loss of generality, we assume that a > b > 0 and let r = (a − b)/(ab) ∈ (0, 1). Then from (1.4) we get

NAQ(a,b)=12A(a,b)[1+(1+r2)tan1(r)r].
3.11

It follows from (3.2), (3.11) and G(a,b)=A(a,b)1r2 that

TD[A(a,b),G(a,b)]G(a,b)NAQ(a,b)G(a,b)2πE(r)1r212[1+(1+r2)tan1(r)r]1r2=[4πrE(r)2r1r2]/(1+r2)tan1(r)+(r2r1r2)/(1+r2).
3.12

Let g1(r)=[4πrE(r)2r1r2]/(1+r2), g2(r)=tan1(r)+(r2r1r2)/(1+r2) and

g(r)=[4πrE(r)2r1r2]/(1+r2)tan1(r)+(r2r1r2)/(1+r2).
3.13

Then simple computations lead to

g1(0+) = g2(0) = 0, 
3.14

g1(r)g2(r)=2π1r2[2ε(r)(1+r2)κ(r)]+3r213r2+1r21=φ4(r)φ5(r),
3.15

where φ4(r) and φ5(r) are defined as in Lemmas 2.6 and 2.7.

It follows from Lemmas 2.6-2.7 and (3.15) that g1(r)/g2(r) is strictly increasing on (0, 1). Then (3.13), (3.14) and Lemma 2.1 lead to the conclusion that g(r) is strictly increasing.

Moreover,

limr0+[4πrε(r)2r1r2]/(1+r2)tan1(r)+(r2r1r2)/(1+r2)=310,
3.16

limr1[4πrε(r)2r1r2]/(1+r2)tan1(r)+(r2r1r2)/(1+r2)=8π(π+2).
3.17

Therefore, Theorem 3.2 follows from (3.12), (3.16) and (3.17) together with the monotonicity of g(r).

From Theorems 3.1-3.2 we get the following Corollary 3.3 immediately.

Corollary 3.3

Let α = 3/8, β=4/[π(log(1+2)+2)]=0.5546 , λ = 3/10 and μ = 8/[π(π + 2)] = 0.4952 ⋯  . Then the double inequalities

14πα[1+r2+sinh1(r)r]+12π(1α)1r2<E(r)<14πβ[1+r2+sinh1(r)r]+12π(1β)1r2,14πλ[1+(1+r2)tan1(r)r]+12π(1λ)1r2<E(r)<14πμ[1+(1+r2)tan1(r)r]+12π(1μ)1r2

hold for all r ∈ (0, 1).

Results and discussion

In this paper, we provide the sharp bounds for the Toader-type mean in terms of the convex combination of geometric and Neuman means. As applications, we find new bounds for the complete elliptic integral of the second kind.

Conclusion

In the article, we present the optimal convex combination bounds of the geometric and Neuman means for the Toader-type mean, and give several new upper and lower bounds for the complete elliptic integral of the second kind. The given results are the improvements of some previously known results.

Acknowledgements

This research was supported by the Natural Science Foundation of Zhejiang Province under Grant LY13A010004 and the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT-15G17.

Footnotes

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Contributor Information

Yue-Ying Yang, moc.361@zh8001yyy.

Wei-Mao Qian, umoc.621@779166mwq.

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