TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 842542 10.1155/2013/842542 842542 Research Article Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean Jiang Wei-Dong Kittaneh F. Sadarangani K. Department of Information Engineering Weihai Vocational College Weihai Shandong 264210 China 2013 22 10 2013 2013 26 08 2013 16 09 2013 2013 Copyright © 2013 Wei-Dong Jiang. 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.

The authors find the greatest value λ and the least value μ, such that the double inequality C¯(λa+(1-λb),λb+(1-λ)a)<αA(a,b)+(1-α)T(a,b)<C¯(μa+(1-μ)b,μb+(1-μ)a) holds for all α(0,1) and a,b>0 with ab, where C¯(a,b)=2(a2+ab+b2)/3(a+b), A(a,b)=(a+b)/2, and Ta,b=2/π0π/2a2cos2θ+b2sin2θdθ denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers a and b.

1. Introduction

In , Toader introduced a mean (1)T(a,b)=2π0π/2a2cos2θ+b2sin2θdθ={2a1-(b/a)2π,a>b,2b1-(a/b)2π,a<b,a,a=b, where (2)=(r)=0π/2(1-r2sin2θ)1/2dθ, for r[0,1] is the complete elliptic integral of the second kind.

In recent years, there have been plenty of literature, such as , dedicated to the Toader mean.

For p and a,b>0, the centroidal mean C¯(a,b) and pth power mean Mp(a,b) are, respectively, defined by (3)C¯(a,b)=2(a2+ab+b2)3(a+b),Mp(a,b)={(ap+ap2)1/p,p0,ab,p=0.

In , Vuorinen conjectured that (4)M3/2(a,b)<T(a,b), for all a,b>0 with ab. This conjecture was verified by Qiu and Shen  and by Barnard et al. , respectively.

In , Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: (5)T(a,b)<Mlog2/log(π/2)(a,b), for all a,b>0 with ab.

Chu et al.  proved that the double inequality (6)C(αa+(1-α)b,αb+(1-α)a)<T(a,b)<C(βa+(1-β)b,βb+(1-β)a) holds for all a,b>0 with ab if and only if α3/4 and β1/2+4π-π2/(2π).

Very recently, Hua and Qi  proved that the double inequality (7)αC¯(a,b)+(1-α)A(a,b)<T(a,b)<βC¯(a,b)+(1-β)A(a,b) is valid for all a,b>0 with ab if and only if α3/4 and β(12/π)-3. Where A(a,b)=(a+b)/2 denote the arithmetic mean.

For positive numbers a,b>0 with ab, let (8)J(x)=C¯(xa+(1-x)b,xb+(1-x)a) be on [1/2,1]. It is not difficult to directly verify that J(x) is continuous and strictly increasing on [1/2,1].

The main purpose of the paper is to find the greatest value λ and the least value μ, such that the double inequality C¯(λa+(1-λb),λb+(1-λ)a)<αA(a,b)+(1-α)T(a,b)<C¯(μa+(1-μ)b,μb+(1-μ)a) holds for all α(0,1) and a,b>0 with ab. As applications, we also present new bounds for the complete elliptic integral of the second kind.

2. Preliminaries and Lemmas

In order to establish our main result, we need several formulas and Lemmas below.

For 0<r<1 and r=1-r2, Legendre’s complete elliptic integrals of the first and second kinds are defined in [12, 13] by (9)𝒦=𝒦(r)=0π/2(1-r2sin2θ)-1/2dθ,𝒦'=𝒦(r)=𝒦(r),𝒦(0)=π2,𝒦(1)=,=(r)=0π/2(1-r2sin2θ)1/2dθ,'=(r)=(r),(0)=π2,(1)=1, respectively.

For 0<r<1, the formulas (10)d𝒦dr=-r2𝒦rr2,ddr=-𝒦r,d(-r2𝒦)dr=r𝒦,d(𝒦-)dr=rr2,(2r1+r)=2-r2𝒦1+r were presented in [14, Appendix E, pages 474-475].

Lemma 1 (see [<xref ref-type="bibr" rid="B3">14</xref>, Theorem 3.21(1), 3.43 exercises 13(a)]).

The function  (-r2𝒦)/r2 is strictly increasing from (0,1) to (π/4,1), and the function 2-r2𝒦 is increasing from (0,1) to (π/2,2).

Lemma 2.

Let u,α(0,1) and (11)fu,α(r)=13ur2-(1-α)(2π(2(r)-(1-r2)𝒦(r))-1). Then, fu,α>0, for all r(0,1) if and only if u3(1-α)(4/π-1), and fu,α<0, for all r(0,1) if and only if u3(1-α)/4.

Proof.

From (11), one has (12)fu,α(0+)=0,(13)fu,α(1-)=13u-(1-α)(4π-1),(14)fu,α(r)=23r[u-3(1-α)g(r)], where g(r)=(1/π)((-r2𝒦)/r2).

We divide the proof into four cases.

Case 1 ( u 3 ( 1 - α ) / π ).  From (14) and Lemma 1 together with the monotonicity of g(r), we clearly see that fu,α(r) is strictly increasing on (0,1). Therefore, fu,α(r)>0, for all r(0,1).

Case 2 ( u 3 ( 1 - α ) / 4 ). From (14) and Lemma 1 together with the monotonicity of g(r), we obtain that fu,α(r) is strictly decreasing on (0,1). Therefore, fu,α(r)<0, for all r(0,1).

Case 3 ( 3 ( 1 - α ) / 4 < u 3 ( 1 - α ) ( 4 / π - 1 ) ). From (13) and (14) together with the monotonicity of g(r), we see that there exists λ(0,1), such that fu,α(r) is strictly increasing in (0,λ] and strictly decreasing in [λ,1) and (15)fu,α(1-)0. Therefore, making use of (12) and inequality (15) together with the piecewise monotonicity of fu,α(r) leads to the conclusion that there exists 0<λ<η<1, such that fu,α(r)>0 for r(0,η) and fu,α(r)<0 for r(η,1).

Case 4 ( 3 ( 1 - α ) ( 4 / π - 1 ) u < 3 ( 1 - α ) / π ). Equation (13) leads to (16)fu,α(1-)0.

From (13) and (14) together with the monotonicity of g(r), we clearly see that there exists λ(0,1), such that fu,α(r) is strictly increasing in (0,λ] and strictly decreasing in [λ,1). Therefore, fu,α(r)>0 for r(0,1) follows from (12) and (16) together with the piecewise monotonicity of fu,α(r).

3. Main Results

Now, we are in a position to state and prove our main results.

Theorem 3.

If α(0,1) and λ,μ(1/2,1), then the double inequality (17)C¯(λa+(1-λ)b,λb+(1-λ)a)<αA(a,b)+(1-α)T(a,b)<C¯(μa+(1-μ)b,μb+(1-μ)a) holds for all a,b>0 with ab if and only if (18)λ12+3(1-α)4,μ12(1+3(1-α)(4π-1)).

Proof.

Since A(a,b), T(a,b), and C¯(a,b) are symmetric and homogeneous of degree one, without loss of generality, we assume that a>b. Let p(1/2,1), t=b/a(0,1), and r=(1-t)/(1+t). Then, (19)C¯(pa+(1-p)b,pb+(1-p)a)-αA(a,b)-(1-α)T(a,b)=a23((p+(1-p)ba)2+(p+(1-p)ba)(pba+1-p)+(pba+1-p)2)(1+ba)-1-αa1+(b/a)2-(1-α)2aπ(1-(ba)2)=a{23((pt+1-p)2(pt+1-p)2(p+(1-p)t)2+(p+(1-p)t)(pt+1-p)+(pt+1-p)2)(1+t)-1-α1+t2-(1-α)2π(1-t2)}=a{(1-2p)2r2+33(1+r)-α11+r-(1-α)2π2-r2𝒦1+r}=a1+r[13(1-2p)2r2+1-α-(1-α)2π(2-r2𝒦)]. Therefore, Theorem 3 follows easily from Lemma 2 and (19).

Let α=1/4,  λ=7/8,  μ=(1/2)(1+(34/π-1/2)). Then, from Theorem 3, we get new bounds for the complete elliptic integral (r) of the second kind in terms of elementary functions as follows.

Corollary 4.

For r(0,1) and r=1-r2, one has (20)π2[5+6r+5r28(1+r)]<(r)<π[r+(2/π)(1-r)21+r].

4. Remarks Remark 5.

In the recent past, the complete elliptic integrals have attracted the attention of numerous mathematicians. In , it was established that (21)π2[121+r22+1+r4]<(r)<π2[4-π(2-1)π1+r22+(2π-4)(1+r)2(2-1)π], for all r(0,1).

Guo and Qi  proved that (22)π2-12log(1+r)1-r(1-r)1+r<(r)<π-12+1-r24rlog1+r1-r, for all r(0,1).

Yin and Qi  presented that (23)π26+21-r2-3r222(r)π210-21-r2-5r222, for all r(0,1).

It was pointed out in  that the bounds in (21) for (r) are better than the bounds in (22) for some r(0,1).

Remark 6.

The lower bound in (20) for (r) is better than the lower bound in (21). Indeed, (24)5+6x+5x28(1+x)-[121+x22+1+x4]=3x2+2x+3-22(1+x2)(1+x)8(1+x),(3x2+2x+3)2-(22(1+x2)(1+x))2=(1-x)4>0, for all x(0,1).

Remark 7.

The following equivalence relations for x(0,1) show that the lower bound in (20) for (r) is better than the lower bound in (23): (25)5+6x+5x28(1+x)>6+2x-3(1-x2)22(5x2+6x+5)2>8(x+1)2(3x2+2x+3)(x-1)4>0.

Acknowledgments

The author is thankful to the anonymous referees for their valuable and profound comments on and suggestions to the original version of this paper. This work was supported by the project of Shandong Higher Education Science and Technology Program under Grant no. J11LA57.

Toader G. Some mean values related to the arithmetic-geometric mean Journal of Mathematical Analysis and Applications 1998 218 2 358 368 2-s2.0-0000408279 10.1006/jmaa.1997.5766 Chu Y.-M. Wang M.-K. Inequalities between arithmetic-geometric, Gini, and Toader means Abstract and Applied Analysis 2012 2012 11 830585 10.1155/2012/830585 Chu Y.-M. Wang M.-K. Optimal lehmer mean bounds for the Toader mean Results in Mathematics 2012 61 3-4 223 229 2-s2.0-78650939191 10.1007/s00025-010-0090-9 Chu Y.-M. Wang M.-K. Qiu S.-L. Optimal combinations bounds of root-square and arithmetic means for Toader mean Proceedings of the Indian Academy of Sciences 2012 122 1 41 51 2-s2.0-84859302835 10.1007/s12044-012-0062-y Chu Y.-M. Wang M.-K. Ma X.-Y. Sharp bounds for Toader mean in terms of contraharmonic mean with applications Journal of Mathematical Inequalities 2013 7 2 161 166 Chu Y.-M. Wang M.-K. Qiu S.-L. Qiu Y.-F. Sharp generalized seiffert mean bounds for toader mean Abstract and Applied Analysis 2011 2011 8 2-s2.0-84555188316 10.1155/2011/605259 605259 Vuorinen M. Hypergeometric functions in geometric function theory Proceedings of the Workshop on Special Functions and Differential Equations January 1998 Madras, India The Institute of Mathematical Sciences 119 126 Qiu S.-L. Shen J.-M. On two problems concerning means Journal of Hangzhou Insitute of Electronic Engineering 1997 17 3 1 7 Barnard R. W. Pearce K. Richards K. C. An inequality involving the generalized hypergeometric function and the arc length of an ellipse SIAM Journal on Mathematical Analysis 2000 31 3 693 699 2-s2.0-0041152123 Alzer H. Qiu S.-L. Monotonicity theorems and inequalities for the complete elliptic integrals Journal of Computational and Applied Mathematics 2004 172 2 289 312 2-s2.0-5044237360 10.1016/j.cam.2004.02.009 Hua Y. Qi F. The best bounds for toader mean in terms of the centroidal and arithmetic mean http://arxiv.org/abs/1303.2451 Bowman F. Introduction to Elliptic Functions with Applications 1961 New York, NY, USA Dover Byrd P. F. Friedman M. D. Handbook of Elliptic Integrals for Engineers and Scientists 1971 New York, NY, USA Springer Anderson G. D. Vamanamurthy M. K. Vuorinen M. Conformal Invariants, Inequalities, and Quasiconformal Maps 1997 New York, NY, USA John Wiley & Sons Guo B.-N. Qi F. Some bounds for the complete elliptic integrals of the first and second kinds Mathematical Inequalities and Applications 2011 14 2 323 334 2-s2.0-79952093367 Yin L. Qi F. Some inequalities for complete elliptic integrals http://arxiv.org/abs/1301.4385