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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 394194, 5 pages
http://dx.doi.org/10.1155/2013/394194
Research Article

Some Inequalities for Bounding Toader Mean

Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin 300387, China

Received 8 May 2013; Revised 11 June 2013; Accepted 16 June 2013

Academic Editor: L. E. Persson

Copyright © 2013 Wen-Hui Li and Miao-Miao Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By finding linear relations among differences between two special means, the authors establish some inequalities for bounding Toader mean in terms of the arithmetic, harmonic, centroidal, and contraharmonic means.

1. Introduction

It is well known that the quantities are, respectively, called in the literature the arithmetic, geometric, harmonic, centroidal, contraharmonic, root-square means, and the power mean of order of two positive numbers and . In [1], Toader introduced a mean where for and is Legendre's complete elliptic integral of the second kind; see [2] and [3, pages 40–46].

In [4], Vuorinen conjectured that for all with . This conjecture was verified in [5, 6], respectively. Later in [7], it was presented that for all with . The constants and which appeared in (4) and (5) are the best possible.

Utilizing inequalities (4) and (5) and using the fact that the power mean is continuous and strictly increasing with respect to for fixed with may conclude that for all with . In [8, Theorem 3.1], it was demonstrated that the double inequality holds for all with if and only if

Recently in [9, Theorems 1.1 to 1.3], it was shown that the double inequalities hold for all with if and only if

The equation (4.4) in [10, page 1013] reads that Motivated by (12), we further find that It is not difficult to see that the double inequality (9) can be rearranged as Therefore, replacing the denominator in (14) by one of differences in (12) and (13) yields where and satisfy (11). On the other hand, the arithmetic mean in the numerator of (14) can also be replaced by the harmonic, contraharmonic, or centroidal means.

For our own convenience, we denote the difference of means in (12) and (13) by The quantity is nonnegative and convex on . See [11, Theorem 2.1].

Now we naturally pose the following problem.

Problem 1. What are the best constants and such that the double inequality holds for all positive numbers and with ?

The main purposes of this paper are to answer the previous problem, to provide an alternative proof for inequalities (14) to (19), and, finally, to remark the connection between Toader mean and the complete elliptic integral of the second kind.

2. Lemmas

To attain our main purposes, we need the following lemmas.

For , denote . Legendre's complete elliptic integrals of the first kind may be defined in [12, 13] by

Lemma 2 (see [14, Appendix , pages 474-475]). For and , one has

Lemma 3 (see [14, Theorem 1.25]). For , let be continuous on , differentiable on , and on . If is (strictly) increasing (or (strictly) decreasing, resp.) on , so are the functions

Lemma 4 (see [14, Theorem 3.21]). The function is strictly increasing and convex from onto .

Lemma 5. The function is strictly decreasing on and satisfies

Proof. By the first three formulas in Lemma 2, simple computations lead to where the function is defined by (25) in Lemma 4.
From it follows that Hence, the function is strictly decreasing on . Further, by easily obtained limits in (27), the proof of Lemma 5 is complete.

3. Some Inequalities for Bounding Toader Mean

Now, we are in a position to give an affirmative solution to Problem 1 and to provide an alternative proof for inequalities (14) to (19).

Theorem 6. The double inequality holds for all with if and only if

Proof. Without loss of generality, assume that . Let . Then, and Let . Then, , and by the last formula in Lemma 2, where By the middle two formulas in Lemma 2, a straightforward calculation leads to So, we have Combining this with Lemmas 3 and 4 reveals that the function is strictly increasing on . Moreover, using L’Hôpital’s rule, we obtain The proof of Theorem 6 is thus complete.

Theorem 7. For all with ,
the double inequality
holds if and only if
the double inequality
holds if and only if
the double inequality
holds if and only if

Proof. From the identities in (12), it follows that Substituting these into the inequality (31) in Theorem 6 acquires inequalities (39) to (43) in Theorem 7.

Theorem 8. The inequality holds for all with if and only if .

Proof. Without loss of generality, assume that . Let . Then, and Let . Then, , and by the last formula in Lemma 2, Therefore, from Lemma 5, it follows that function is strictly decreasing and Theorem 8 is thus proved.

4. Remarks

Finally, we would like to remark several things, including the connection between Toader mean and the complete elliptic integral of the second kind.

Remark 9. The double inequality (39) is equivalent to (9). Consequently, inequalities (14) to (19) are recovered once again.

Remark 10. The coefficient in (12) corrects an error which appeared at the corresponding position in (4.4) in [10, page 1013]. Luckily, this error does not influence the correctness of any other conclusions in [10].

Remark 11. We point out that Toader mean satisfies where is the complete symmetric elliptic integral of the second kind and is a symmetric and homogeneous function; see [15, equation (9.2-3)] and [16, page 250, equation (11)]. Numerous inequalities involving , , and are known in the mathematical literature; see [2, 1619] and [3, pages 40–46] and closely related references therein. In the past years, the fact that Toader mean and the elliptic integral are the same has been overlooked by several researchers.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments on the original version of this paper and Professor Feng Qi for his kind help in the whole process of composing this paper.

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