Journal of Function Spaces

Volume 2016, Article ID 3698463, 6 pages

http://dx.doi.org/10.1155/2016/3698463

## Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete Elliptic Integral

^{1}Department of Mathematics and System Science, National University of Defense Technology, Changsha 410073, China^{2}Department of Mathematics, Changsha University of Science and Technology, Changsha 410014, China^{3}School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China^{4}School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

Received 8 May 2016; Revised 11 June 2016; Accepted 23 June 2016

Academic Editor: Rudolf L. Stens

Copyright © 2016 Hua Wang et al. 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.

#### Abstract

We present the best possible parameters and such that the double inequalities , , hold for all with , where , , and are the arithmetic, quadratic, and Gauss arithmetic-geometric means of and , respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.

#### 1. Introduction

Let and . Then the elliptic elliptic integral of the first kind and second kind , Gaussian arithmetic-geometric mean , arithmetic mean , and quadratic mean are, respectively, given by

The Gauss identity [1–3] shows that for all , where and in what follows .

It is well known that the elliptic elliptic integrals and and the Gaussian arithmetic-geometric mean have many applications in mathematics, physics, mechanics, and engineering [4–9]. Recently, the bounds for the Gaussian arithmetic-geometric mean have attracted the attention of many researchers.

The inequalities for all and with can be found in the literature [10–12], where and are, respectively, the logarithmic and th generalized logarithmic means of and . The first inequality of (5) is due to Carlson and Vuorinen [13].

By using a variant of L’Hospital’s rule and representation theorems with elliptic integrals, Vamanamurthy and Vuorinen [14] proved, among other results, the inequalities for all with and , where is the identric mean of and .

By use of the homogeneity of the above means and a series representation of due to Gauss, Sándor [15] obtained, among other results, new proofs for inequalities (7), (8) and a counterpart of inequality (9): for all with , where is the geometric mean of and . Inequalities (9) and (12) show that lies between the arithmetic and geometric means of and . In [16], Sándor provided new proofs for inequalities (6) and (8), (9), (10), and (12) by using only elementary methods for recurrent sequences and found much stronger forms of these results.

Neuman and Sándor [17] gave the comparison of the Gaussian arithmetic-geometric mean and the Schwab-Borchardt mean.

The upper bounds for in (4) were replaced by due to Kühnau [18].

Qiu and Vamanamurthy [19] presented that and are, respectively, the lower and upper bounds for with . Alzer and Qiu [20] proved that and are the best possible parameters such that the double inequality holds for all with .

Chu and Wang [21] proved that the double inequality holds for all with if and only if and , where and is the th Gini mean of and . In [22], Yang et al. proved that the inequalitieshold for all and with , where is the Stolarsky mean [23] of and .

Let with and . Then it is not difficult to verify that the function is continuous and strictly increasing on the interval . Note that

Inequalities (16) give us the motivation to deal with the best possible parameters and such that the double inequalities hold for all with .

#### 2. Lemmas

In order to prove our main results we need several derivative formulas and particular values for and , which we present in this section.

and satisfy the formulas (see [24])where is the classical Euler gamma function.

Lemma 1 (see [24, Theorem 1.25]). *Let , be continuous on and differentiable on and on . Then both functions are increasing (decreasing) on if is increasing (decreasing) on . If is strictly monotone, then the monotonicity in the conclusion is also strict.*

Lemma 2 (see [24, Theorem 3.21(1), Theorem 3.21(7), and Exercises 3.43(32)]). *The following statements are true:*(1)*The function is strictly increasing from onto .*(2)*The function is strictly decreasing from onto if .*(3)*The function is strictly increasing from onto .*

*Lemma 3. Let and be defined by Then there exists such that for and for .*

*Proof. *From (21) we clearly see that can be rewritten as It follows from Lemma 2 and together with (22) that is strictly increasing on .

Numerical computations show that Therefore, Lemma 3 follows easily from (23) and the monotonicity of on the interval .

*3. Main Results*

*Theorem 4. The double inequality holds for all with if and only if and .*

*Proof. *Since , , and are symmetric and homogenous of degree 1, without loss of generality, we assume that . Let . Then (2) and (3) lead to Let Then simple computations give It follows from Lemmas 1, 2 and (27) and (28) that and is strictly increasing on the interval .

Note that Therefore, Theorem 4 follows easily from (26), (27), (29), and (30) and the monotonicity of on the interval .

*Remark 5. *The left side inequality of Theorem 4 for can be derived directly from the fact that and for all with .

*Theorem 6. The double inequality holds for all with if and only if and .*

*Proof. *Without loss of generality, we assume that . Let . Then it follows from (2) and (3) that Let Then simple computations lead to It follows from Lemmas 1, 2(1) and (2) together with (33) and (34) that and is strictly decreasing on the interval .

Note that Therefore, Theorem 6 follows easily from (32), (33), (35), and (36) and the monotonicity of on the interval .

*Remark 7. *The right side inequality of Theorem 6 for can be derived directly from the fact that and for all with .

*Theorem 8. Let . Then the double inequality holds for all with if and only if and .*

*Proof. *Without loss of generality, we assume that . Let and . Then (2) and (3) lead to wherewhere is defined by (21).

We divide the proof into four cases.*Case 1 *(). Then (42) becomesIt follows from Lemma 3 and (43) that there exists such that is strictly decreasing on and strictly increasing on . Therefore, follows from (39), (41), and (44) together with the piecewise monotonicity of on the interval .*Case 2 *(). Then we clearly see that *Case 3 *(). Then (42) leads to Equation (39) and inequality (47) imply that there exists small enough such that for all with .*Case 4 *(). Then (40) leads to Note that Equations (39) and (49) together with inequality (50) imply that there exists small enough such that for all with .

*4. Applications*

*In this section, we use Theorems 4, 6, and 8 to present several bounds for the complete elliptic integrals and .*

*From Theorems 4, 6, and 8 we get Theorem 9 immediately.*

*Theorem 9. Let , , and . Then the double inequalities hold for all .*

*It follows from the inequality given in [24] that *

*Theorem 9 and (54) lead to the following.*

*Theorem 10. Let , , and . Then the double inequalities hold for all .*

* Competing Interests *

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*The research was supported by the Natural Science Foundation of China under Grants 11371125, 61374086, and 11401191, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, and the Natural Science Foundation of the Zhejiang Broadcast and TV University under Grant XKT-15G17.*

*References*

- J. M. Borwein and P. B. Borwein, “The arithmetic-geometric mean and fast computation of elementary functions,”
*SIAM Review*, vol. 26, no. 3, pp. 351–366, 1984. View at Publisher · View at Google Scholar · View at MathSciNet - J. M. Borwein and P. B. Borwein,
*Pi and the AGM*, John Wiley & Sons, New York, NY, USA, 1987. - G. Almkvist and B. Berndt, “Gauss, landen, ramanujan, the arithmetic-geometric mean, ellipses, $\pi $ and the ladies diary,”
*The American Mathematical Monthly*, vol. 95, no. 7, pp. 585–608, 1988. View at Google Scholar - B. C. Carlson, “Hidden symmetries of special functions,”
*SIAM Review*, vol. 12, pp. 332–345, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - I. Elishakoff, V. Birman, and J. Singer, “Influence of initial imperfections on nonlinear free vibration of elastic bars,”
*Acta Mechanica*, vol. 55, no. 3-4, pp. 191–202, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. K. Lee, “Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds,”
*Computer Physics Communications*, vol. 60, no. 3, pp. 319–327, 1990. View at Publisher · View at Google Scholar - T. Horiguchi, “Lattice Green's function for anisotropic triangular lattice,”
*Physica A*, vol. 178, no. 2, pp. 351–363, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - K. Mayrhofer and F. D. Fischer, “Derivation of a new analytical solution for a general two-dimensional finite-part integral applicable in fracture mechanics,”
*International Journal for Numerical Methods in Engineering*, vol. 33, no. 5, pp. 1027–1047, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. C. Maican,
*Integral Evaluations Using the Gamma and Beta Functions and Elliptic Integrals in Engineering*, International Press, Cambridge, Mass, USA, 2005. - J. M. Borwein and P. B. Borwein, “Inequalities for compound mean iterations with logarithmic asymptotes,”
*Journal of Mathematical Analysis and Applications*, vol. 177, no. 2, pp. 572–582, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - P. Bracken, “An arithmetic-geometric mean inequality,”
*Expositiones Mathematicae*, vol. 19, no. 3, pp. 273–279, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Zh.-H. Yang, “A new proof of inequalities for Gauss compound mean,”
*International Journal of Mathematical Analysis*, vol. 4, no. 21–24, pp. 1013–1018, 2010. View at Google Scholar · View at MathSciNet · View at Scopus - B. C. Carlson and M. Vuorinen, “Problem 91-17,”
*SIAM Review*, vol. 33, no. 4, p. 655, 1991. View at Google Scholar - M. K. Vamanamurthy and M. Vuorinen, “Inequalities for means,”
*Journal of Mathematical Analysis and Applications*, vol. 183, no. 1, pp. 155–166, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Sándor, “On certain inequalities for means,”
*Journal of Mathematical Analysis and Applications*, vol. 189, no. 2, pp. 602–606, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Sándor, “On certain inequalities for means II,”
*Journal of Mathematical Analysis and Applications*, vol. 199, no. 2, pp. 629–635, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - E. Neuman and J. Sándor, “On the Schwab-Borchardt mean,”
*Mathematica Pannonica*, vol. 14, no. 2, pp. 253–266, 2003. View at Google Scholar · View at MathSciNet - R. Kühnau, “Eine Methode, die Positivität einer Funktion zu prüfen,”
*Zeitschrift für Angewandte Mathematik und Mechanik*, vol. 74, no. 2, pp. 140–143, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - S.-L. Qiu and M. K. Vamanamurthy, “Sharp estimates for complete elliptic integrals,”
*SIAM Journal on Mathematical Analysis*, vol. 27, no. 3, pp. 823–834, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Alzer and S.-L. Qiu, “Monotonicity theorems and inequalities for the complete elliptic integrals,”
*Journal of Computational and Applied Mathematics*, vol. 172, no. 2, pp. 289–312, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y.-M. Chu and M.-K. Wang, “Inequalities between arithmetic-geometric, Gini, and Toader means,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 830585, 11 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Z.-H. Yang, Y.-Q. Song, and Y.-M. Chu, “Sharp bounds for the arithmetic-geometric mean,”
*Journal of Inequalities and Applications*, vol. 2014, article 192, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Zh.-H. Yang, Y.-M. Chu, and W. Zhang, “Accurate approximations for the complete elliptic integral of the second kind,”
*Journal of Mathematical Analysis and Applications*, vol. 438, no. 2, pp. 875–888, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen,
*Conformal Invariants, Inequalities, and Quasiconformal Maps*, John Wiley & Sons, New York, NY, USA, 1997.

*
*