Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

Chu, Yu-Ming; Wu, Li-Min; Song, Ying-Qing

doi:https://doi.org/10.1155/2015/517647

Journal of Function Spaces

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Abstract Introduction Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 517647 | https://doi.org/10.1155/2015/517647

Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

Yu-Ming Chu,¹Li-Min Wu,²and Ying-Qing Song¹

Academic Editor: David R. Larson

Received03 Nov 2014

Accepted27 Apr 2015

Published30 Apr 2015

Abstract

We present the best possible parameters and such that the double inequality holds for all and with , where and and are the power and one-parameter harmonic means of and , respectively.

1. Introduction

For and , the th power mean of and is defined by

It is well known that is strictly increasing with respect to for fixed with , symmetric and homogeneous of degree 1. Many classical means are special cases of the power mean: for example, is the harmonic mean, is the geometric mean, is the arithmetic mean, and is the quadratic mean. The main properties of the power mean are given in [1]. Recently, the power mean has attracted the attention of many researchers. In particular, many remarkable inequalities for the power mean can be found in the literature [2–10].

Let , , , , and be the logarithmic, first Seiffert, identric, second Seiffert, and contraharmonic means of two distinct positive real numbers and , respectively. Then it is well known that the inequalities hold for all with .

Lin [11] proved that the double inequality holds for all with if and only if and .

In [12], Pittenger presented the best possible parameters and such that the double inequality holds for all with , where , , and is the generalized logarithmic mean of and .

Jagers [13] and Seiffert [14] proved that the double inequalities hold for all with .

In [15, 16], the authors proved that the double inequalities

hold for all with .

Costin and Toader [17] proved that the double inequality holds for all with .

In [18–20], the authors proved that the double inequalities hold for all with if and only if , , , and .

Čizmesija [21] proved that and are the best possible parameters such that the double inequality holds for all and with .

In [22, 23], the authors proved that the inequalitieshold for all if and only if , , and , where and are, respectively, the complete elliptic integrals of the first and second kinds.

Let and be the bivariate symmetric mean. Then, the one-parameter mean was defined by Neuman [24] as follows:

Let and . Then, the authors in [25–28] proved that the inequalities hold for all with if and only if , , , , , , , , , , and , where is the unique solution of the equation .

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

2. Lemmas

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

Lemma 1. The inequality holds for all .

Proof. It is not difficult to verify that for all . Therefore, we only need to prove that for , where . Simple computations lead to where for all .
Inequality (15) implies that for all . Then, from (13) we clearly see that for all .

Lemma 2. The inequality holds for all .

Proof. Let and . Then, it is not difficult to verify that for all .
It follows from Lemma 1 thatfor all .
Inequalities (17) and (18) lead to for all .

Lemma 3. The inequality holds for all .

Proof. Let and . Then, it follows from (17) and (18) that for all .

3. Main Results

Theorem 4. The double inequality holds for all and with if and only if and .

Proof. Without loss of generality, we assume that and . Let , where Then, simple computations lead towhere whereWe divide the proof into two cases.
Case 1 (). We divide the discussion into two subcases.
Subcase 1.1 ( and ). Then, we clearly see that , and (28), (29), (32), and (33) lead to for .
It follows easily from (24), (25), (27), (30), and (34)–(36) that for all .
Subcase 1.2 ( and ). Then, , and (37) follows from Case 2 (). Then, we clearly see that , and Lemmas 1–3 and (23), (26), (28), (29), and (32) lead to and (36) again holds.
It follows from (30), (36), and (45) that is strictly increasing on . Then, (43) and (44) lead to the conclusion that there exists such that is strictly decreasing on and strictly increasing on .
From (41) and (42) together with the piecewise monotonicity of we clearly see that there exists such that is strictly decreasing on and strictly increasing on . Then, (25), (27), and (40) lead to the conclusion that there exists such that is strictly decreasing on and strictly increasing on . Therefore, for all follows from (24) and (39) together with the piecewise monotonicity of .
Next, we prove that and are the best possible parameters such that the double inequality holds for all and with .
Let , , , , and . Then, we have Let and make use of the Taylor expansion; then, (48) leads to Inequality (49) and equation (50) imply that for any and there exist and such that for and for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the Natural Science Foundation of China under Grants 61374086 and 11371125, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, and the Natural Science Foundation of Hunan Province under Grant 14JJ2127.

References

P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities, D. Reidel Publishing Company, Dordrecht, The Netherlands, 1988.
View at: MathSciNet
C. O. Imoru, “The power mean and the logarithmic mean,” International Journal of Mathematics and Mathematical Sciences, vol. 5, no. 2, pp. 337–343, 1982.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. E. Pečarić, “Generalization of the power means and their inequalities,” Journal of Mathematical Analysis and Applications, vol. 161, no. 2, pp. 395–404, 1991.
View at: Publisher Site | Google Scholar | MathSciNet
D. Lukkassen, “Means of power type and their inequalities,” Mathematische Nachrichten, vol. 205, pp. 131–147, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z. Liu, “Remark on inequalities between Hölder and Lehmer means,” Journal of Mathematical Analysis and Applications, vol. 247, no. 1, pp. 309–313, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
H. Alzer, “A power mean inequality for the gamma function,” Monatshefte für Mathematik, vol. 131, no. 3, pp. 179–188, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
C. Mortici, “Arithmetic mean of values and value at mean of arguments for convex functions,” The ANZIAM Journal, vol. 50, no. 1, pp. 137–141, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
C. Mortici, “A power series approach to some inequalities,” The American Mathematical Monthly, vol. 119, no. 2, pp. 147–151, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
L.-M. Zhou, S.-L. Qiu, and F. Wang, “Inequalities for the generalized elliptic integrals with respect to Hölder means,” Journal of Mathematical Analysis and Applications, vol. 386, no. 2, pp. 641–646, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
B. A. Bhayo, “On the power mean inequality of the hyperbolic metric of unit ball,” The Journal of Prime Research in Mathematics, vol. 8, pp. 45–50, 2012.
View at: Google Scholar | MathSciNet
T. P. Lin, “The power mean and the logarithmic mean,” The American Mathematical Monthly, vol. 81, pp. 879–883, 1974.
View at: Publisher Site | Google Scholar | MathSciNet
A. O. Pittenger, “Inequalities between arithmetic and logarithmic means,” Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, vol. 678–715, pp. 15–18, 1980.
View at: Google Scholar
A. A. Jagers, “Solutions of problem 887,” Nieuw Archief voor Wiskunde, vol. 12, no. 4, pp. 230–231, 1994.
View at: Google Scholar
H. J. Seiffert, “Aufgabe $β$ 16,” Die Wurzel, vol. 29, pp. 221–222, 1995.
View at: Google Scholar
H. Alzer, “Ungleichungen für Mittelwerte,” Archiv der Mathematik, vol. 47, no. 5, pp. 422–426, 1986.
View at: Publisher Site | Google Scholar | MathSciNet
H. Alzer and S.-L. Qiu, “Inequalities for means in two variables,” Archiv der Mathematik, vol. 80, no. 2, pp. 201–215, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
I. Costin and Gh. Toader, “A separation of some Seiffert-type means by power means,” Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 41, no. 2, pp. 125–129, 2012.
View at: Google Scholar | MathSciNet
P. A. Hastö, “Optimal inequalities between Seiffert's mean and power means,” Mathematical Inequalities & Applications, vol. 7, no. 1, pp. 47–53, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
I. Costin and Gh. Toader, “Optimal evaluations of some Seiffert-type means by power means,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4745–4754, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-M. Li, M.-K. Wang, and Y.-M. Chu, “Sharp power mean bounds for Seiffert mean,” Applied Mathematics, vol. 29, no. 1, pp. 101–107, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
A. Čizmesija, “A new sharp double inequality for generalized Heronian, harmonic and power means,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 664–671, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
M.-K. Wang, Y.-M. Chu, Y.-F. Qiu, and S.-L. Qiu, “An optimal power mean inequality for the complete elliptic integrals,” Applied Mathematics Letters, vol. 24, no. 6, pp. 887–890, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-M. Chu, S.-L. Qiu, and M.-K. Wang, “Sharp inequalities involving the power mean and complete elliptic integral of the first kind,” The Rocky Mountain Journal of Mathematics, vol. 43, no. 5, pp. 1489–1496, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
E. Neuman, “A one-parameter family of bivariate means,” Journal of Mathematical Inequalities, vol. 7, no. 3, pp. 399–412, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-M. Chu, M.-K. Wang, and Z.-K. Wang, “A best-possible double inequality between Seiffert and harmonic means,” Journal of Inequalities and Applications, vol. 2011, article 94, 7 pages, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-M. Chu, M.-K. Wang, and Z.-K. Wang, “Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means,” Mathematical Inequalities & Applications, vol. 15, no. 2, pp. 415–422, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
Y.-M. Chu and S.-W. Hou, “Sharp bounds for Seiffert mean in terms of contraharmonic mean,” Abstract and Applied Analysis, vol. 2012, Article ID 425175, 6 pages, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
M.-K. Wang, Z.-K. Wang, and Y.-M. Chu, “An optimal double inequality between geometric and identric means,” Applied Mathematics Letters, vol. 25, no. 3, pp. 471–475, 2012.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2015 Yu-Ming Chu et al. 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.

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