1. Introduction

For the th Gini mean and power mean of two positive real numbers and are defined by respectively.

It is well known that and are continuous and strictly increasing with respect to for fixed with . Many means are special case of these means, for example,

Recently, the Gini and power means have been the subject of intensive research. In particular, many remarkable inequalities for these means can be found in the literature [1–7].

In [8], Toader introduced the Toader mean of two positive numbers and as follows: where , , is the complete elliptic integrals of the second kind.

The classical arithmetic-geometric mean of two positive number and is defined as the common limit of sequences and , which are given by

The Gauss identity [9] shows that for , where , , is the complete elliptic integrals of the first kind.

Vuorinen [10] conjectured that for all with . This conjecture was proved by Qiu and Shen in [11] and Barnard et al. in [12], respectively.

In [13], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: for all with .

In [14–17], the authors proved that for all with , where denotes the classical logarithmic mean of two positive numbers and .

Very recently, Chu and Wang [18] and Guo and Qi [19] proved that for all with , and and are the best possible lower and upper Lehmer mean bounds for the Toader mean , respectively. Here, the th Lehmer mean of two positive numbers and is defined by .

The main purpose of this paper is to find the greatest values , and least values , such that the double inequalities and hold for all with and present some new bounds for the complete elliptic integrals.

2. Preliminary Knowledge

Throughout this paper, we denote for .

For , the following derivative formulas were presented in [9, Appendix E, pages 474–475]:

Lemma 2.1 can be found in [9, Theorem  3.21(7), (8), and (10), and Exercise  3.43(13) and (46)].

Lemma 2.1. (1) is strictly decreasing from onto for ;
(2)  is strictly increasing on if and only if and strictly decreasing if and only if ;
(3) is strictly decreasing from onto ;
(4) is strictly increasing from onto ;
(5) is strictly decreasing from onto .

3. Main Results

Theorem 3.1. Inequality holds for all with , and and are the best possible lower and upper Gini mean bounds for the arithmetic-geometric mean .

Proof. From (1.1) and (1.5) we clearly see that both and are symmetric and homogenous of degree 1. Without loss of generality, we assume that . Let and . Then from (1.1) and (1.6) together with (2.2) we clearly see that
Let Then can be rewritten as where
It is well known that the function is positive and strictly decreasing in . Then (3.4) and Lemma 2.1(1) lead to the conclusion that is strictly increasing in , so that for .
Therefore, follows from (3.1)–(3.3).
On the other hand, follows directly from (1.9).
Next, we prove that and are the best possible lower and upper Gini mean bounds for the arithmetic-geometric mean .
For any and , from (1.1), (1.6), and Lemma 2.1(3) we have Letting and making use of the Taylor expansion, one has
Equations (3.5)–(3.7) imply that for any there exist and , such that for and for .

Theorem 3.2. Inequality holds for all with , and and are the best possible lower and upper Gini mean bounds for the Toader mean .

Proof. From (1.1) and (1.4) we clearly see that both and are symmetric and homogenous of degree 1. Without loss of generality, we assume that . Let and . Then from (1.1), (1.4), and (2.3) we have
Let Then simple computations lead to where
It follows from (3.12) and Lemma 2.1(1), (4), and (5) that is strictly decreasing from onto . Then (3.11) leads to the conclusion that for . Hence is strictly decreasing in .
Therefore, follows from (3.8)–(3.10) together with the monotonicity of .
On the other hand, follows directly from (1.7).
Next, we prove that and are the best possible lower and upper Gini mean bounds for the Toader mean .
For any and , from (1.1) and (1.4) one has
Letting and making use of the Taylor expansion, we get
Equations (3.13)–(3.15) imply that for any   there exist and , such that for and for .

4. Remarks and Corollaries

Remark 4.1. From (3.9) and Lemma 2.1(4) we clearly see that . Then (3.8) and (3.9) together with the monotonicity of lead to the conclusion that for all with .

Remark 4.2. We find that the lower bound in (1.10) and the best possible lower Gini mean bound in Theorem 3.1 are not comparable. In fact, from (1.1) and (1.11) we have

Remark 4.3. The following two equations show that the best possible upper power mean bound in (1.8) and the best possible upper Gini mean bound in Theorem 3.2 are not comparable: