Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2010, Article ID 303286, 13 pages
http://dx.doi.org/10.1155/2010/303286
Research Article

Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2College of Mathematics Science, Anhui University, Hefei 230039, China

Received 26 November 2009; Revised 16 February 2010; Accepted 16 March 2010

Academic Editor: Lance Littlejohn

Copyright © 2010 Yu-Ming Chu 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 answer the question: for with , what are the greatest value and the least value , such that the double inequality holds for all with ? Here , , , and denote the generalized logarithmic, arithmetic, geometric, and harmonic means of two positive numbers and , respectively.

1. Introduction

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

It is well known that is continuous and strictly increasing with respect to for fixed with . Recently, the generalized logarithmic mean has been the subject of intensive research. Many remarkable inequalities and monotonicity results for the generalized logarithmic mean can be found in the literature [19]. It might be surprising that the generalized logarithmic mean has applications in physics, economics, and even in meteorology [1013].

Let , , , , and be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers and with , respectively. Then

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

In [14], Alzer and Janous established the following sharp double inequality (see also [15, page 350]):for all with .

For , Janous [16] found the greatest value and the least value such thatfor all with .

In [1719], the authors presented bounds for and in terms of and .

Theorem A. For all positive real numbers and with , we have

The proof of the following Theorem B can be found in [20].

Theorem B. For all positive real numbers and with , we have

The following Theorems C–E were established by Alzer and Qiu in [21].

Theorem C. The inequalitieshold for all positive real numbers and with if and only if and .

Theorem D. Let and be real numbers with . If , thenAnd if , then

Theorem E. For all positive real numbers and with , we havewith the best possible parameter .

However, the following problem is still open: for with , what are the greatest value and the least value , such that the double inequalityholds for all with ? The purpose of this paper is to give the solution to this open problem.

2. Lemmas

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

Lemma 2.1. If , then

Proof. If , then it follows from [22, ] thatIt is not difficult to verify thatfor .
Therefore, Lemma 2.1 follows from inequalities (2.2) and (2.3).

Lemma 2.2. If , then

Proof. Let . Then simple computations lead towhere ,where ,for .
Therefore, Lemma 2.2 follows from (2.5)–(2.7), (2.9)–(2.11), (2.13), and (2.14).

Lemma 2.3. Let and +. Then(1) for and(2) for .

Proof. Simple computations yieldwhere +,

() If , then (2.22) implies

for .

Therefore, Lemma 2.3() follows from (2.15), (2.17), (2.19), (2.20), (2.21), and (2.23).

() If , then (2.22) leads tofor .

Therefore, Lemma 2.3() follows from (2.15), (2.17), (2.19), (2.20), (2.21), and (2.24).

Lemma 2.4. Let and . Then(1) for and(2) for .

Proof. From the expression of , we clearly see thatFrom (2.28), we havefor and , andfor and .
Therefore, Lemma 2.4() follows from (2.25) and (2.27) together with (2.29), and Lemma 2.4() follows from (2.25) and (2.27) together with (2.30).

3. Main Result

Theorem 3.1. Let with and with . Then(1) for ;(2) for and for , and the parameters and cannot be improved in either case.

Proof. () We divide the proof into two cases.
Case  1. If , then simple computations lead toCase  2. If , then we clearly see that() Without loss of generality, we assume that and we put and .
Firstly, we compare with . We divide the proof into five cases.
Case  1. If , then (1.1) leads toTherefore, follows from (3.3) and Lemma 2.1.
Case  2. If , then from (1.1) we haveFrom (3.4) and Lemma 2.2, we clearly see that .
Case  3. If , then (1.1) yieldsLet . Then simple computations lead towhere ++.
We clearly see thatTherefore, follows from (3.5)–(3.8) and Lemma 2.3().
Case  4. If , then (3.5)–(3.8) again hold, and from (3.5)–(3.8) and Lemma 2.3(), we know that
Case  5. If , then (3.5)–(3.7) hold andTherefore, follows from (3.5)–(3.7) and (3.9) together with Lemma 2.3().
Secondly, we compare with .
From (1.1), we haveLet . Then simple computations yieldwhere
We clearly see thatfor .
Therefore, for follows from (3.10)–(3.13) and Lemma 2.4(), and for follows from (3.10)–(3.13) and Lemma 2.4().
At last, we prove that the parameters and cannot be improved in either case.
The following two cases will complete the proof for the optimality of parameter .
Case I. If , then for any , one hasEquation (3.14) implies that for any , there exists a sufficiently large , such that for .
Case II. If , then for any , one hasEquation (3.15) implies that for any , there exists a sufficiently large , such that for .
The following seven cases will complete the proof for the optimality of parameter .
Case A. If , then for any and , one haswhere
Upon letting , the Taylor expansion leads toEquations (3.16) and (3.17) imply that for , there exists a sufficiently small , such that for .
Case B. If , then for any and , one hasEquation (3.18) implies that for any , there exists a sufficiently small , such that for .
Case C. If , then for any and , one hasEquation (3.19) implies that for any , there exists a sufficiently small , such that for .
Case D. If , then for any and , one hasEquation (3.20) implies that for any , there exists a sufficiently small , such that for .

Case E. If , then for any and , one has

Equation (3.21) implies that for any , there exists a sufficiently small , such that for .

Case F. If , then for any and , one has

Equation (3.22) implies that for any , there exists a sufficiently small , such that for .

Case G. If , then for any and , one has

Equation (3.23) implies that for any , there exists a sufficiently small , such that for .

Acknowledgments

The authors wish to thank the anonymous referee for his very careful reading of the manuscript and fruitful comments and suggestions. This research is partly supported by N S Foundation of China (Grant 60850005), N S Foundation of Zhejiang Province (Grants D7080080 and Y607128), and Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant T200924).

References

  1. K. B. Stolarsky, “The power and generalized logarithmic means,” The American Mathematical Monthly, vol. 87, no. 7, pp. 545–548, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. C. E. M. Pearce and J. Pečarić, “Some theorems of Jensen type for generalized logarithmic means,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 40, no. 9-10, pp. 789–795, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. B. Mond, C. E. M. Pearce, and J. Pečarić, “The logarithmic mean is a mean,” Mathematical Communications, vol. 2, no. 1, pp. 35–39, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Ch.-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. 499, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. A. K. Abuhany, S. R. Salem, and I. M. Salman, “On Steffensen's integral inequality with applications,” Journal of Rajasthan Academy of Physical Sciences, vol. 5, no. 1, pp. 1–12, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. F. Qi, Sh.-X. Chen, and Ch.-P. Chen, “Monotonicity of ratio between the generalized logarithmic means,” Mathematical Inequalities & Applications, vol. 10, no. 3, pp. 559–564, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. H.-N. Shi, “Schur-convex functions related to hadamard-type inequalities,” Journal of Mathematical Inequalities, vol. 1, no. 1, pp. 127–136, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Ch.-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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. F. Qi, X.-A. Li, and Sh.-X. Chen, “Refinements, extensions and generalizations of the second Kershaw's double inequality,” Mathematical Inequalities & Applications, vol. 11, no. 3, pp. 457–465, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. O. Pittenger, “The logarithmic mean in n variables,” The American Mathematical Monthly, vol. 92, no. 2, pp. 99–104, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. N. S. Nadirashvili, “New isoperimetric inequalities in mathematical physics,” in Partial Differential Equations of Elliptic Type (Cortona, 1992), Sympos. Math., XXXV, pp. 197–203, Cambridge University Press, Cambridge, UK, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University, Princeton, NJ, USA, 1951. View at MathSciNet
  14. H. Alzer and W. Janous, “Solution of problem 8,” Crux Mathematicorum with Mathematical Mayhem, vol. 13, pp. 173–178, 1987. View at Google Scholar
  15. P. S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities, vol. 31, D. Reidel, Dordrecht, The Netherlands, 1988. View at MathSciNet
  16. W. Janous, “A note on generalized heronian means,” Mathematical Inequalities & Applications, vol. 4, no. 3, pp. 369–375, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. B. C. Carlson, “The logarithmic mean,” The American Mathematical Monthly, vol. 79, pp. 615–618, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. B. E. Leach and M. C. Sholander, “Extended mean values II,” Journal of Mathematical Analysis and Applications, vol. 92, no. 1, pp. 207–223, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Sándor, “A note on some inequalities for means,” Archiv der Mathematik, vol. 56, no. 5, pp. 471–473, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. H. Alzer, “Ungleichungen für mittelwerte,” Archiv der Mathematik, vol. 47, no. 5, pp. 422–426, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. D. S. Mitrinović, Analytic Inequalities, Springer, New York, NY, USA, 1970. View at MathSciNet