Optimal convex combination bounds of geometric and Neuman means for Toader-type mean.

In this paper, we prove that the double inequalities [Formula: see text] hold for all [Formula: see text] with [Formula: see text] if and only if [Formula: see text], [Formula: see text] , [Formula: see text] and [Formula: see text] , where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively.

Introduction

For with and , the symmetric integrals and [1] of the first and second kinds, and the complete elliptic integrals and of the first and second kinds are defined by respectively.

The well-known identities were established by Carlson in [1].

Let with . Then the Toader mean [2] and the Schwab-Borchardt mean [3-5] are respectively defined by and where and are the inverse cosine and inverse hyperbolic cosine functions, respectively.

Very recently, Neuman [6] introduced the Neuman mean of the second kind as follows:

It is well known that the Toader mean , the Schwab-Borchardt mean and the Neuman mean of the second kind satisfy the identities (see [6, 7])

Let and . Then the pth power mean is defined by

We clearly see that is symmetric and homogeneous of degree one with respect to a and b, strictly increasing with respect to for fixed with , and the inequalities hold for with , where , and are the geometric, arithmetic and quadratic means of a and b, respectively.

In [6], Neuman presented the explicit formula for and as follows: and proved that the inequalities hold for with , where .

Recently, the Toader mean has been the subject of intensive research. In particular, many remarkable inequalities for Toader mean and other related means can be found in the literature [8-41].

In [42], Vuorinen conjectured that for all with . This conjecture was proved by Qiu and Shen [43], and Barnard et al. [44], respectively, and Alzer and Qiu [45] presented the best possible upper power mean bound for the Toader mean as follows: for all with .

Li, Qian and Chu [46] proved that the inequality holds for all with if and only if and  .

Note that for all with .

From inequalities (1.5) and (1.6) we clearly see that for all with .

The main purpose of this paper is to find the greatest values α, λ and the least values β, μ such that the double inequalities hold for all with . As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions.

Lemmas

In order to prove our main results, we need several lemmas, which we present in this section.

For , we clearly see that and and satisfy the formulas (see[21], Appendix E, pp.474-475)

Lemma 2.1

see [21], Theorem 1.25

For , let be continuous on and differentiable on , and on . If is increasing (decreasing) on , then so are

If is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2.2

see [21], Theorem 3.21(1), Exercise 3.43(11) and Exercise 3.43(29)

The function is strictly increasing from onto ;

The function is strictly increasing from onto ;

The function is strictly increasing from onto .

Lemma 2.3

The function is strictly increasing from onto .

Proof

Simple computations lead to where

From (2.5) and Lemma 2.2(3) we get

Therefore, Lemma 2.3 follows easily from (2.1), (2.2), (2.4) and (2.6). □

Lemma 2.4

The function is strictly decreasing from onto .

It is easy to verify that for .

Therefore, Lemma 2.4 follows easily from (2.7) and (2.8). □

Lemma 2.5

The function is strictly increasing from onto .

It is not difficult to verify that

From (2.10) and Lemma 2.2(2) together with the monotonicity of on we clearly see that for .

Therefore, Lemma 2.5 follows from (2.9) and (2.11). □

Lemma 2.6

The function is strictly increasing from onto .

Let , . Then simple computations give

It follows from Lemma 2.2(1), Lemma 2.5 and the function strictly decreasing that is strictly increasing on and

Therefore, Lemma 2.6 follows from Lemma 2.1, (2.12), (2.13) and (2.15) together with the monotonicity of . □

Lemma 2.7

The function is strictly decreasing from onto .

We clearly see that for .

Therefore, Lemma 2.7 follows easily from (2.16) and (2.17). □

Main results

Theorem 3.1

The double inequality holds for all with if and only if and  .

Since , and are symmetric and homogenous of degree 1, without loss of generality, we assume that and let . Then (1.1)-(1.3) lead to

It follows from (3.2)-(3.3) that

Let , and

Then simple computations lead to where and are defined as in Lemmas 2.3 and 2.4.

It follows from Lemmas 2.3-2.4 and (3.7) that is strictly increasing on . Then (3.5), (3.6) and Lemma 2.1 lead to the conclusion that is strictly increasing.

Moreover,

Therefore, Theorem 3.1 follows easily from (3.4), (3.8) and (3.9) together with the monotonicity of . □

Theorem 3.2

The double inequality holds for all with if and only if and  .

Without loss of generality, we assume that and let . Then from (1.4) we get

It follows from (3.2), (3.11) and that

Let , and

Then simple computations lead to where and are defined as in Lemmas 2.6 and 2.7.

It follows from Lemmas 2.6-2.7 and (3.15) that is strictly increasing on . Then (3.13), (3.14) and Lemma 2.1 lead to the conclusion that is strictly increasing.

Moreover,

Therefore, Theorem 3.2 follows from (3.12), (3.16) and (3.17) together with the monotonicity of . □

From Theorems 3.1-3.2 we get the following Corollary 3.3 immediately.

Corollary 3.3

Let ,  , and  . Then the double inequalities hold for all .

Results and discussion

In this paper, we provide the sharp bounds for the Toader-type mean in terms of the convex combination of geometric and Neuman means. As applications, we find new bounds for the complete elliptic integral of the second kind.

Conclusion

In the article, we present the optimal convex combination bounds of the geometric and Neuman means for the Toader-type mean, and give several new upper and lower bounds for the complete elliptic integral of the second kind. The given results are the improvements of some previously known results.