Journal of Function Spaces
Volume 2016 (2016), Article ID 3698463, 6 pages
http://dx.doi.org/10.1155/2016/3698463

Research Article

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

Hua Wang,1,2 Wei-Mao Qian,³ and Yu-Ming Chu⁴

¹Department of Mathematics and System Science, National University of Defense Technology, Changsha 410073, China
²Department of Mathematics, Changsha University of Science and Technology, Changsha 410014, China
³School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China
⁴School 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 inequalities

hold 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

where

where is defined by (21).
We divide the proof into four cases.
Case  1 (). Then (42) becomes

It 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.

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