Skip to content


  • Research Article
  • Open Access

Proof of One Optimal Inequality for Generalized Logarithmic, Arithmetic, and Geometric Means

Journal of Inequalities and Applications20102010:902432

  • Received: 11 July 2010
  • Accepted: 31 October 2010
  • Published:


Two open problems were posed in the work of Long and Chu (2010). In this paper, we give the solutions of these problems.


  • Real Number
  • Unique Solution
  • Open Problem
  • Simple Computation
  • Elementary Calculation

1. Introduction

The arithmetic and geometric means of two positive numbers and are defined by , , respectively. If is a real number, then the generalized logarithmic mean with parameter of two positive numbers , is defined by

In the paper [1], Long and Chu propose the two following open problems:

Open Problem 1.

What is the least value such that the inequality

holds for and all with

Open Problem 2.

What is the greatest value such that the inequality

holds for and all with

For information on the history, background, properties, and applications of inequalities for generalized logarithmic, arithmetic, and geometric means, please refer to [119] and related references there in.

The aim of this article is to prove the following Theorem 2.1.

2. Main Result

Theorem 2.1.

Let , , , . Let be a solution of
and is the best constant,

and is the best constant.

3. Proof of Theorem 2.1

Because is increasing with respect to for fixed and , it suffices to prove that for any (resp., ) there exists such that (resp., ), and is the best constant. Without loss of generality, we assume that . Let , . Equations (2.2), (2.3) are equivalent to
On putting , we obtain (3.1) is equivalent to
Introduce the function by
Simple computations yield for
Let and the unique solution to
To see that is optimal in both cases (2.2), (2.3), note that . Thus, if the constant is decreased (resp., increased), then the desired bound for would not hold for small . This follows from the fact that for a fixed , the function

is nondecreasing.

From now on, let for . To show the estimates for this , we start from observing that . Furthermore, one easily checks that
Thus, it suffices to verify that has exactly one zero inside the interval . It follows from the mean value theorem. After some computations, this is equivalent to saying that the function given by

has exactly one root in . Here, the expression under the logarithm may be nonpositive, so we define on a maximal interval, contained in . It is easy to see that this interval must be of the form , for some . This follows from the fact that is strictly positive on and is strictly increasing on this interval.

Since and , we will be done if we show that has exactly one root in . After some computations, we obtain that the equation is equivalent to
Because is a quadratic polynomial in the variable , all that remains is to show that
or, in virtue of the definition of ,

This can be easily established by some elementary calculations. It completes the proof.



The author is indebted to the anonymous referee for many valuable comments, for a correction of one part of the proof, and for his improving of the organization of the paper. This work was supported by Vega no. 1/0157/08 and Kega no. 3/7414/09.

Authors’ Affiliations

Faculty of Industrial Technologies in Púchov, Alexander Dubček University in Trenčín, I. Krasku 491/30, 02001 Púchov, Slovakia


  1. Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. Journal of Inequalities and Applications 2010, 2010:-10.MathSciNetMATHGoogle Scholar
  2. Alzer H: Ungleichungen für Mittelwerte. Archiv der Mathematik 1986, 47(5):422–426. 10.1007/BF01189983MathSciNetView ArticleMATHGoogle Scholar
  3. Alzer H, Qiu S-L: Inequalities for means in two variables. Archiv der Mathematik 2003, 80(2):201–215. 10.1007/s00013-003-0456-2MathSciNetView ArticleMATHGoogle Scholar
  4. Burk F: The geometric, logarithmic, and arithmetic mean inequality. The American Mathematical Monthly 1987, 94(6):527–528. 10.2307/2322844MathSciNetView ArticleMATHGoogle Scholar
  5. Janous W: A note on generalized Heronian means. Mathematical Inequalities & Applications 2001, 4(3):369–375.MathSciNetView ArticleMATHGoogle Scholar
  6. Leach EB, Sholander MC: Extended mean values. II. Journal of Mathematical Analysis and Applications 1983, 92(1):207–223. 10.1016/0022-247X(83)90280-9MathSciNetView ArticleMATHGoogle Scholar
  7. Sándor J: On certain inequalities for means. Journal of Mathematical Analysis and Applications 1995, 189(2):602–606. 10.1006/jmaa.1995.1038MathSciNetView ArticleMATHGoogle Scholar
  8. Sándor J: On certain inequalities for means. II. Journal of Mathematical Analysis and Applications 1996, 199(2):629–635. 10.1006/jmaa.1996.0165MathSciNetView ArticleMATHGoogle Scholar
  9. Sándor J: On certain inequalities for means. III. Archiv der Mathematik 2001, 76(1):34–40. 10.1007/s000130050539MathSciNetView ArticleMATHGoogle Scholar
  10. Shi M-Y, Chu Y-M, Jiang Y-P: Optimal inequalities among various means of two arguments. Abstract and Applied Analysis 2009, 2009:-10.MathSciNetView ArticleMATHGoogle Scholar
  11. Carlson BC: The logarithmic mean. The American Mathematical Monthly 1972, 79: 615–618. 10.2307/2317088MathSciNetView ArticleMATHGoogle Scholar
  12. Sándor J: On the identric and logarithmic means. Aequationes Mathematicae 1990, 40(2–3):261–270.MathSciNetView ArticleMATHGoogle Scholar
  13. Sándor J: A note on some inequalities for means. Archiv der Mathematik 1991, 56(5):471–473. 10.1007/BF01200091MathSciNetView ArticleMATHGoogle Scholar
  14. Lin TP: The power mean and the logarithmic mean. The American Mathematical Monthly 1974, 81: 879–883. 10.2307/2319447MathSciNetView ArticleMATHGoogle Scholar
  15. Pittenger AO: Inequalities between arithmetic and logarithmic means. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta 1980, (678–715):15–18.MathSciNetMATHGoogle Scholar
  16. Imoru CO: The power mean and the logarithmic mean. International Journal of Mathematics and Mathematical Sciences 1982, 5(2):337–343. 10.1155/S0161171282000313MathSciNetView ArticleMATHGoogle Scholar
  17. Chen C-P: The monotonicity of the ratio between generalized logarithmic means. Journal of Mathematical Analysis and Applications 2008, 345(1):86–89. 10.1016/j.jmaa.2008.03.071MathSciNetView ArticleMATHGoogle Scholar
  18. Li X, Chen C-P, Qi F: Monotonicity result for generalized logarithmic means. Tamkang Journal of Mathematics 2007, 38(2):177–181.MathSciNetMATHGoogle Scholar
  19. Qi F, Chen S-X, Chen C-P: Monotonicity of ratio between the generalized logarithmic means. Mathematical Inequalities & Applications 2007, 10(3):559–564.MathSciNetView ArticleMATHGoogle Scholar


© Ladislav Matejíčka. 2010

This article is published under license to BioMed Central Ltd. 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.