Journal of Applied MathematicsVolume 2012 (2012), Article ID 480689, 8 pagesdoi:10.1155/2012/480689

Research Article

Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means

Wei-Mao Qian¹ and Bo-Yong Long²

¹School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China²School of Mathematics Science, Anhui University, Hefei 230039, China

Received 29 January 2012; Revised 19 February 2012; Accepted 12 March 2012

Academic Editor: Yuri Sotskov

Copyright © 2012 Wei-Mao Qian and Bo-Yong Long. 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.

We present sharp upper and lower generalized logarithmic mean bounds for the geometric weighted mean of the geometric and harmonic means.

1. Introduction

For the generalized logarithmic mean of two positive numbers and is defined by

It is well-known that is continuous and strictly increasing with respect to for fixed and with . In the recent past, the generalized logarithmic mean has been the subject of intensive research. In particular, many remarkable inequalities for can be found in the literature [1–23]. The generalized logarithmic mean has applications in convex function, economics, physics, and even in meteorology [24–27]. In [26] the authors study a variant of Jensen’s functional equation involving , which appear in a heat conduction problem. Let , , , and be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers and with , respectively. Then it is well known that

In [28–30], the authors present bounds for and in terms of and .

Proposition 1.1. For all positive real numbers and with , one has

The proof of the following Proposition 1.2 can be found in [31].

Proposition 1.2. For all positive real numbers and with , we have

For the th power mean of two positive numbers and is defined by

The main properties of these means are given in [32]. Several authors discussed the relationship of certain means to . The following sharp bounds for , , , and in terms of power means are proved in [31, 33–37].

Proposition 1.3. For all positive real numbers and with one has

The following three results were established by Alzer and Qiu in [38].

Proposition 1.4. The inequalities hold for all positive real numbers and with if and only if

Proposition 1.5. Let and be real numbers with . If , then And, if , then

Proposition 1.6. For all positive real numbers and with , one has with the best possible parameter

In [39] the authors presented inequalities between the generalized logarithmic mean and the product for all with and with .

It is the aim of this paper to give a solution to the problem: for , what are the greatest value and the least value , such that the inequality holds for all ?

2. Main Result

Theorem 2.1. For and all , one has the following: (1) for ,(2) for , and for , with equality if and only if , and the parameters and in each inequality cannot be improved.

Proof. (1) If and , then (1.1) implies that .If and , then (1.1) leads to (2) If , then from (1.1) we clearly see that for any . If , without loss of generality, we assume . Let and Then (1.1) and simple computations yield where , If , then (2.7) implies for .From (2.3)–(2.6) and (2.8) we know that for .If , then (2.7) leads to for . Therefore for follows from (2.3)–(2.6) and (2.9).Let for ; then (1.1) and elementary calculations lead to where , If , then (2.15) implies for .From (2.11)–(2.14) and (2.16) we know that for .If , then (2.15) leads to for . Therefore, for follows from (2.11)–(2.14) and (2.17).Next, we prove that the parameters and in either case cannot be improved. The proof is divided into two cases.Case 1 (). For any and , from (1.1) one has where .Let ; making use of the Taylor expansion, we get Equations (2.18) and (2.19) imply that for any and there exists , such that for .On the other hand, for any we have From (2.20) we know that for any and there exists , such that for .Case 2 (). For any and , from (1.1) one has where .Let ; making use of the Taylor expansion, we have Equations (2.21) and (2.22) imply that for any and there exists , such that for .On the other hand, for any , we have From (2.23) we know that for any and there exists , such that for .

Acknowledgment

This work was supported by the Natural Science Foundation of Zhejiang Broad-cast and TV University under Grant XKT-09G21.

References

K. B. Stolarsky, “The power and generalized logarithmic means,” The American Mathematical Monthly, vol. 87, no. 7, pp. 545–548, 1980.

F. Qi and B.-N. Guo, “An inequality between ratio of the extended logarithmic means and ratio of the exponential means,” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 229–237, 2003.

C.-P. Chen and F. Qi, “Monotonicity properties for generalized logarithmic means,” The Australian Journal of Mathematical Analysis and Applications, vol. 1, no. 2, article 2, p. 4, 2004.

X. Li, C.-P. Chen, and F. Qi, “Monotonicity result for generalized logarithmic means,” Tamkang Journal of Mathematics, vol. 38, no. 2, pp. 177–181, 2007.

F. Qi, S.-X. Chen, and C.-P. Chen, “Monotonicity of ratio between the generalized logarithmic means,” Mathematical Inequalities & Applications, vol. 10, no. 3, pp. 559–564, 2007.

C.-P. Chen, “The monotonicity of the ratio between generalized logarithmic means,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 86–89, 2008.

B.-N. Guo and F. Qi, “A simple proof of logarithmic convexity of extended mean values,” Numerical Algorithms, vol. 52, no. 1, pp. 89–92, 2009.

Y.-M. Chu and W.-F. Xia, “Inequalities for generalized logarithmic means,” Journal of Inequalities and Applications, vol. 2009, Article ID 763252, 7 pages, 2009.

M.-Y. Shi, Y.-M. Chu, and Y.-P. Jiang, “Optimal inequalities among various means of two arguments,” Abstract and Applied Analysis, vol. 2009, Article ID 694394, 10 pages, 2009.

W.-F. Xia and Y.-M. Chu, “Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means,” Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 39, no. 2, pp. 176–183, 2010.

Y.-M. Chu and W.-F. Xia, “Two optimal double inequalities between power mean and logarithmic mean,” Computers & Mathematics with Applications, vol. 60, no. 1, pp. 83–89, 2010.

W.-F. Xia, Y.-M. Chu, and G.-D. Wang, “The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means,” Abstract and Applied Analysis, vol. 2010, Article ID 604804, 9 pages, 2010.

B.-Y. Long and Y.-M. Chu, “Optimal inequalities for generalized logarithmic, arithmetic, and geometric means,” Journal of Inequalities and Applications, vol. 2010, Article ID 806825, 10 pages, 2010.

Y.-M. Chu, S.-S. Wang, and C. Zong, “Optimal lower power mean bound for the convex combination of harmonic and logarithmic means,” Abstract and Applied Analysis, vol. 2011, Article ID 520648, 9 pages, 2011.

M.-Y. Shi, Y.-M. Chu, and Y.-P. Jiang, “Optimal inequalities related to the power, harmonic and identric means,” Acta Mathematica Scientia A, vol. 31, no. 5, pp. 1377–1384, 2011 (Chinese).

Y.-F. Qiu, M.-K. Wang, Y.-M. Chu, and G.-D. Wang, “Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean,” Journal of Mathematical Inequalities, vol. 5, no. 3, pp. 301–306, 2011.

Y.-M. Chu, M.-K. Wang, and Z.-K. Wang, “A sharp double inequality between harmonic and identric means,” Abstract and Applied Analysis, vol. 2011, Article ID 657935, 7 pages, 2011.

Y.-M. Chu and M.-K. Wang, “Optimal inequalities between harmonic, geometric, logarithmic, and arithmetic-geometric means,” Journal of Applied Mathematics, vol. 2011, Article ID 618929, 9 pages, 2011.

Y.-M. Chu, S.-W. Hou, and W.-M. Gong, “Inequalities between logarithmic, harmonic, arithmetic and centroidal means,” SIAM Journal on Mathematical Analysis, vol. 2, no. 2, pp. 1–5, 2011.

H.-N. Hu, S.-S. Wang, and Y.-M. Chu, “Optimal upper power mean bound for the convex combiantion of harmonic and logarithmic means,” Pacific Journal of Applied Mathematics, vol. 4, no. 1, pp. 35–44, 2011.

Y.-F. Qiu, M.-K. Wang, and Y.-M. Chu, “The sharp combination bounds of arithmetic and logarithmic means for Seiffert's mean,” International Journal of Pure and Applied Mathematics, vol. 72, no. 1, pp. 11–18, 2011.

Y.-F. Qiu, M.-K. Wang, and Y.-M. Chu, “The optimal generalized Heronian mean bounds for the identric mean,” International Journal of Pure and Applied Mathematics, vol. 72, no. 1, pp. 19–26, 2011.

M.-K. Wang, Z.-K. Wang, and Y.-M. Chu, “An optimal double inequality between geometric and identric means,” Applied Mathematics Letters, vol. 25, pp. 471–475, 2012.

B.-N. Guo and F. Qi, “Inequalities for generalized weighted mean values of convex function,” Mathematical Inequalities & Applications, vol. 4, no. 2, pp. 195–202, 2001.

A. O. Pittenger, “The logarithmic mean in n variables,” The American Mathematical Monthly, vol. 92, no. 2, pp. 99–104, 1985.

P. Kahlig and J. Matkowski, “Functional equations involving the logarithmic mean,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 76, no. 7, pp. 385–390, 1996.

G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, NJ, USA, 1951.

E. B. Leach and M. C. Sholander, “Extended mean values. II,” Journal of Mathematical Analysis and Applications, vol. 92, no. 1, pp. 207–223, 1983.

J. Sándor, “A note on some inequalities for means,” Archiv der Mathematik, vol. 56, no. 5, pp. 471–473, 1991.

B. C. Carlson, “The logarithmic mean,” The American Mathematical Monthly, vol. 79, pp. 615–618, 1972.

H. Alzer, “Ungleichungen für Mittelwerte,” Archiv der Mathematik, vol. 47, no. 5, pp. 422–426, 1986.

P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities, vol. 31, D. Reidel, Dordrecht, The Netherlands, 1988.

H. Alzer, “Ungleichungen fur ,” Elemente der Mathematik, vol. 40, no. 5, pp. 120–123, 1985.

F. Burk, “Notes: the geometric, logarithmic, and arithmetic mean inequality,” The American Mathematical Monthly, vol. 94, no. 6, pp. 527–528, 1987.

T. P. Lin, “The power mean and the logarithmic mean,” The American Mathematical Monthly, vol. 81, pp. 879–883, 1974.

A. O. Pittenger, “Inequalities between arithmetic and logarithmic means,” Univerzitet u Beogradu, vol. 678–715, pp. 15–18, 1980.

A. O. Pittenger, “The symmetric, logarithmic and power means,” Univerzitet u Beogradu, vol. 678–715, pp. 19–23, 1980.

H. Alzer and S.-L. Qiu, “Inequalities for means in two variables,” Archiv der Mathematik, vol. 80, no. 2, pp. 201–215, 2003.

Y.-M. Chu and B.-Y. Long, “Best possible inequalities between generalized logarithmic mean and classical means,” Abstract and Applied Analysis, vol. 2010, Article ID 303286, 13 pages, 2010.