Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters.

In the article, we present the best possible parameters [Formula: see text] and [Formula: see text] on the interval [Formula: see text] such that the double inequality [Formula: see text] holds for any [Formula: see text] and all [Formula: see text] with [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively.

Introduction

Let . Then the Legendre complete elliptic integrals and [1, 2] of the first and second kinds are defined as respectively. It is well known that the function is strictly increasing from onto and the function is strictly decreasing from onto , and they satisfy the formulas (see [3, Appendix E, pp. 474,475]) where .

The complete elliptic integrals and are the particular cases of the Gaussian hypergeometric function [4-10] where for , is the shifted factorial function and () is the gamma function [11-18]. Indeed,

Recently, the bounds for the complete elliptic integrals have attracted the attention of many researchers. In particular, many remarkable inequalities and properties for , and can be found in the literature [19-52].

In 1998, a class of quasi-arithmetic mean was introduced by Toader [53] which is defined by where for , , and p is a strictly monotonic function. It is well known that many important means are the special cases of the quasi-arithmetic mean. For example, is the arithmetic-geometric mean of Gauss [54-60], is the Toader mean [61-70], and is the Toader-Qi mean [71-74].

Let and . Then reduces to a special quasi-arithmetic mean

Let be the arithmetic, geometric and pth power means of a and b, respectively. Then it is well known that the inequality holds for all with , and the double inequality holds for all (see [75, 19.9.4]).

From (1.1)-(1.3) we clearly see that for all with .

Let and Then it is not difficult to verify that the function is strictly increasing on for fixed and with . Note that for all and with .

Motivated by inequalities (1.4) and the monotonicity of the function on the interval , in the article, we shall find the best possible parameters on the interval such that the double inequality holds for any and all with .

Lemmas

Lemma 2.1

(see [3, Theorem 1.25])

Let , be continuous on and differentiable on , and on . If is increasing (decreasing) on , then so are the functions If is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2.2

The inequality holds for all .

Proof

Let Then simple computations lead to for .

Therefore, Lemma 2.2 follows easily from (2.1)-(2.3). □

Lemma 2.3

The following statements are true:

The function is strictly increasing from onto .

The function is strictly increasing from onto .

The function is strictly increasing from onto .

The function is strictly decreasing from onto .

The function is strictly decreasing from onto .

Parts (1) and (2) can be found in the literature [3, Theorem 3.21(1) and Exercise 3.43(11)].

For part (3), let . Then simple computations lead to It follows from part (1) and (2.5) that for all . Therefore, part (3) follows from (2.4) and (2.6).

For part (4), let , then one has From part (1) and (2.8) we clearly see that for all . Therefore, part (4) follows from (2.7) and (2.9).

For part (5), let , then can be rewritten as Therefore, part (5) follows easily from parts (2) and (4) together with (2.10). □

Lemma 2.4

The function is strictly decreasing from onto .

Let and . Then we clearly see that

From Lemma 2.3(3), (2.11) and (2.13) we know that and the function is strictly decreasing on .

Therefore, Lemma 2.4 follows easily from Lemma 2.1, (2.11), (2.12) and (2.14) together with the monotonicity of the function . □

Lemma 2.5

Let , , and Then one has

for all if and only if ;

for all if and only if .

It follows from (2.15) that where where and are defined by (2.10) and Lemma 2.4, respectively.

From Lemma 2.3(5) and Lemma 2.4 together with (2.19) we clearly see that the function is strictly increasing on and From Lemma 2.2 we know that . Therefore, we only need to divide the proof into three cases as follows.

Case 1 . Then Lemma 2.3(4), (2.18), (2.20) and the monotonicity of the function on the interval lead to the conclusion that the function is strictly increasing on . Therefore, for all follows from (2.16) and the monotonicity of the function .

Case 2 . Then from Lemma 2.2, Lemma 2.3(5), (2.17), (2.18), (2.20), (2.21) and the monotonicity of the function on the interval we clearly see that there exists such that the function is strictly decreasing on and strictly increasing on , and Therefore, for all follows from (2.16) and (2.22) together with the piecewise monotonicity of the function on the interval .

Case 3 . Then (2.17) leads to It follows from Lemma 2.3(5), (2.18), (2.20), (2.21) and the monotonicity of the function on the interval that there exists such that the function is strictly decreasing on and strictly increasing on . Therefore, there exists such that for and for . □

Main result

Theorem 3.1

Let . Then the double inequality holds for any and all with if and only if and .

Let , since and are symmetric and homogeneous of degree one, without loss of generality, we assume that . Let and . Then (1.1) leads to Therefore, Theorem 3.1 follows easily from Lemma 2.5 and (3.1). □

Let , then Theorem 3.1 leads to Corollary 3.2 immediately.

Corollary 3.2

Let . Then the double inequalities hold for all with if and only if  ,  , and .

Let , , , , and . Then (1.1) and Theorem 3.1 lead to Corollary 3.3 immediately.

Corollary 3.3

The double inequality holds for all and .

Results and discussion

In this paper, we provide the sharp bounds for the special quasi-arithmetic mean in terms of the arithmetic mean and geometric mean with two parameters. As consequences, we present the best possible one-parameter harmonic and geometric means bounds for and find new bounds for the complete elliptic integral of the second kind.

Conclusion

In the article, we derive a new bivariate mean from the quasi-arithmetic mean and provide its sharp upper and lower bounds in terms of the concave combination of arithmetic and geometric means.