Optimal inequalities for bounding Toader mean by arithmetic and quadratic means.

In this paper, we present the best possible parameters [Formula: see text] and [Formula: see text] such that the double inequality [Formula: see text] holds for all [Formula: see text] and [Formula: see text] with [Formula: see text], and we provide new bounds for the complete elliptic integral [Formula: see text] [Formula: see text] of the second kind, where [Formula: see text], [Formula: see text] and [Formula: see text] are the Toader, arithmetic, and quadratic means of a and b, respectively.

Introduction

For , and with , the pth generalized Seiffert mean , qth Gini mean , qth power mean , qth Lehmer mean , harmonic mean , geometric mean , arithmetic mean , quadratic mean , Toader mean [1], centroidal mean , contraharmonic mean are, respectively, defined by

It is well known that , , , and are continuous and strictly increasing with respect to and for fixed with , and the inequalities hold for all with .

The Toader mean has been well known in the mathematical literature for many years, it satisfies where stands for the symmetric complete elliptic integral of the second kind (see [2-4]), therefore it cannot be expressed in terms of the elementary transcendental functions.

Let , and be, respectively, the complete elliptic integrals of the first and second kind. Then , , and satisfy the derivatives formulas (see [5], Appendix E, p.474-475) the values and can be expressed as (see [6], Theorem 1.7) where is the Euler gamma function, and the Toader mean can be rewritten as

Recently, the Toader mean has been the subject of intensive research. Vuorinen [7] conjectured that the inequality holds for all with . This conjecture was proved by Qiu and Shen [8], and Barnard, Pearce and Richards [9], respectively.

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

Neuman [2], and Kazi and Neuman [3] proved that the inequalities hold for all with , where is the arithmetic-geometric mean of a and b.

In [11-13], the authors presented the best possible parameters and such that the double inequalities , and hold for all with .

Let . Then Chu, Wang and Ma [14], and Hua and Qi [15] proved that the double inequalities hold for all with if and only if , , and .

In [16-20], the authors proved that the double inequalities hold for all with if and only if , , , , , , , , , , , , , , , , , , , and .

The main purpose of this paper is to present the best possible parameters and such that the double inequality holds for all and with .

Lemmas

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

Lemma 2.1

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

Proof

It follows from (2.1) that where for all .

It follows from (2.13) that is strictly decreasing on .

We divide the proof into three cases.

Case 1 . Then (2.11) leads to

From (2.14) and the monotonicity of we clearly see that is strictly decreasing on . Therefore, for all follows easily from (2.2), (2.4), (2.5), (2.8), and the monotonicity of .

Case 2 . Then from (2.11) and (2.12) together with we clearly see that

It follows from (2.15) and the monotonicity of that there exists such that is strictly increasing on and strictly decreasing on .

Let and  . We divide the proof into three subcases.

Subcase 2.1 . Then (2.9) leads to

It follows from (2.8) and (2.16) together with the piecewise monotonicity of that for all .

Therefore, for all follows easily from (2.2), (2.4), (2.5), and (2.17).

Subcase 2.2 . Then (2.6) and (2.9) lead to

It follows from (2.8) and (2.19) together with the piecewise monotonicity of that there exists such that is strictly increasing on and strictly decreasing on .

Equation (2.5) and inequality (2.18) together with the piecewise monotonicity of lead to the conclusion that for all .

Therefore, for all follows easily from (2.2) and (2.4) together with (2.20).

Subcase 2.3 . Then (2.3), (2.6), and (2.9) lead to

It follows from (2.8) and (2.23) together with the piecewise monotonicity of that there exists such that is strictly increasing on and strictly decreasing on .

From (2.4), (2.5), and (2.22) together with the piecewise monotonicity of we clearly see that there exists such that is strictly increasing on and strictly decreasing on .

Therefore, for all follows easily from (2.2) and (2.21) together with the piecewise monotonicity of .

Case 3 . Then (2.3), (2.6), (2.9), (2.11), and (2.12) lead to

It follows from (2.27) and (2.28) together with the monotonicity of that there exists such that is strictly increasing on and strictly decreasing on .

Equation (2.8) and inequality (2.26) together with the piecewise monotonicity of lead to the conclusion that there exists such that is strictly increasing on and strictly decreasing on .

From (2.4), (2.5), (2.25), and the piecewise monotonicity of we clearly see that there exists such that is strictly increasing on and strictly decreasing on .

Therefore, there exists such that for and for follows from (2.2) and (2.24) together with the piecewise monotonicity of . □

Lemma 2.2

Let , with ,  , , and be defined by and respectively. Then the function is strictly decreasing on .

Let , , and Then from (2.29)-(2.31) one has where inequalities (2.35) and (2.36) hold due to and the function is strictly decreasing on .

Note that and the function is strictly increasing on . Then (2.34)-(2.36) lead to the conclusion that there exists such that the function is strictly decreasing on and strictly increasing on .

It follows from (2.32) and (2.33) together with the piecewise monotonicity of the function on the interval that for all with .

Therefore, Lemma 2.2 follows from (2.37). □

Main result

Theorem 3.1

Let  , and be defined by (2.29). Then the double inequality holds for all and with if and only if and , where is the limit value of .

We first prove that Theorem 3.1 holds for .

Since for all with , and , and are symmetric and homogeneous of degree 1, without loss of generality, we assume that , and . Let and . Then (1.1) and (1.2) lead to where is defined as in Lemma 2.1.

Therefore, Theorem 3.1 for follows easily from Lemma 2.1 and (3.1).

Next, let and with , then it follows from Theorem 3.1 for that

Note that

Therefore, Theorem 3.1 for follows from (3.2)-(3.6) and Lemma 2.2 together with the monotonicity of the function . □

Let . Then Theorems 3.1 leads to Corollary 3.2 immediately.

Corollary 3.2

Let . Then the double inequality holds for all .