1. Introduction

In [1], Toader introduced a mean where for is the complete elliptic integral of the second kind.

In recent years, there have been plenty of literature, such as [2–6], dedicated to the Toader mean.

For and , the centroidal mean and th power mean are, respectively, defined by

In [7], Vuorinen conjectured that for all with . This conjecture was verified by Qiu and Shen [8] and by Barnard et al. [9], respectively.

In [10], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: for all with .

Chu et al. [5] proved that the double inequality holds for all with if and only if and .

Very recently, Hua and Qi [11] proved that the double inequality is valid for all with if and only if and . Where denote the arithmetic mean.

For positive numbers with , let be on . It is not difficult to directly verify that is continuous and strictly increasing on .

The main purpose of the paper is to find the greatest value and the least value , such that the double inequality holds for all and with . 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 and , Legendre’s complete elliptic integrals of the first and second kinds are defined in [12, 13] by respectively.

For , the formulas were presented in [14, Appendix E, pages 474-475].

Lemma 1 (see [14, Theorem 3.21(1), 3.43 exercises 13(a)]). The function   is strictly increasing from to , and the function is increasing from to .

Lemma 2. Let and Then, , for all if and only if , and , for all if and only if .

Proof. From (11), one has where .
We divide the proof into four cases.
Case 1 ().  From (14) and Lemma 1 together with the monotonicity of , we clearly see that is strictly increasing on . Therefore, , for all .
Case 2 (). From (14) and Lemma 1 together with the monotonicity of , we obtain that is strictly decreasing on . Therefore, , for all .
Case 3 (). From (13) and (14) together with the monotonicity of , we see that there exists , such that is strictly increasing in and strictly decreasing in and Therefore, making use of (12) and inequality (15) together with the piecewise monotonicity of leads to the conclusion that there exists , such that for and for .
Case 4 (). Equation (13) leads to
From (13) and (14) together with the monotonicity of , we clearly see that there exists , such that is strictly increasing in and strictly decreasing in . Therefore, for follows from (12) and (16) together with the piecewise monotonicity of .

3. Main Results

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

Theorem 3. If and , then the double inequality holds for all with if and only if

Proof. Since , , and are symmetric and homogeneous of degree one, without loss of generality, we assume that . Let , , and . Then, Therefore, Theorem 3 follows easily from Lemma 2 and (19).

Let ,  ,  . Then, from Theorem 3, we get new bounds for the complete elliptic integral of the second kind in terms of elementary functions as follows.

Corollary 4. For and , one has

4. Remarks

Remark 5. In the recent past, the complete elliptic integrals have attracted the attention of numerous mathematicians. In [4], it was established that for all .
Guo and Qi [15] proved that for all .
Yin and Qi [16] presented that for all .
It was pointed out in [4] that the bounds in (21) for are better than the bounds in (22) for some .

Remark 6. The lower bound in (20) for is better than the lower bound in (21). Indeed, for all .

Remark 7. The following equivalence relations for show that the lower bound in (20) for is better than the lower bound in (23):

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.