• Research Article
• Open Access

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

Journal of Inequalities and Applications20102010:902432

https://doi.org/10.1155/2010/902432

• Accepted: 31 October 2010
• Published:

Abstract

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

Keywords

• 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
(1.1)

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
(1.2)

holds for and all with

Open Problem 2.

What is the greatest value such that the inequality
(1.3)

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.

2. Main Result

Theorem 2.1.

Let , , , . Let be a solution of
(2.1)
Then,
(2.2)
and is the best constant,
(2.3)

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
(3.1)
On putting , we obtain (3.1) is equivalent to
(3.2)
Introduce the function by
(3.3)
Simple computations yield for
(3.4)
Let and the unique solution to
(3.5)
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
(3.6)

is nondecreasing.

From now on, let for . To show the estimates for this , we start from observing that . Furthermore, one easily checks that
(3.7)
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
(3.8)

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
(3.9)
Because is a quadratic polynomial in the variable , all that remains is to show that
(3.10)
or, in virtue of the definition of ,
(3.11)

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

Declarations

Acknowledgments

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

(1)
Faculty of Industrial Technologies in Púchov, Alexander Dubček University in Trenčín, I. Krasku 491/30, 02001 Púchov, Slovakia

References

1. Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. Journal of Inequalities and Applications 2010, 2010:-10.
2. Alzer H: Ungleichungen für Mittelwerte. Archiv der Mathematik 1986, 47(5):422–426. 10.1007/BF01189983
3. Alzer H, Qiu S-L: Inequalities for means in two variables. Archiv der Mathematik 2003, 80(2):201–215. 10.1007/s00013-003-0456-2
4. Burk F: The geometric, logarithmic, and arithmetic mean inequality. The American Mathematical Monthly 1987, 94(6):527–528. 10.2307/2322844
5. Janous W: A note on generalized Heronian means. Mathematical Inequalities & Applications 2001, 4(3):369–375.
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)90280-9
7. Sándor J: On certain inequalities for means. Journal of Mathematical Analysis and Applications 1995, 189(2):602–606. 10.1006/jmaa.1995.1038
8. Sándor J: On certain inequalities for means. II. Journal of Mathematical Analysis and Applications 1996, 199(2):629–635. 10.1006/jmaa.1996.0165
9. Sándor J: On certain inequalities for means. III. Archiv der Mathematik 2001, 76(1):34–40. 10.1007/s000130050539
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.
11. Carlson BC: The logarithmic mean. The American Mathematical Monthly 1972, 79: 615–618. 10.2307/2317088
12. Sándor J: On the identric and logarithmic means. Aequationes Mathematicae 1990, 40(2–3):261–270.
13. Sándor J: A note on some inequalities for means. Archiv der Mathematik 1991, 56(5):471–473. 10.1007/BF01200091
14. Lin TP: The power mean and the logarithmic mean. The American Mathematical Monthly 1974, 81: 879–883. 10.2307/2319447
15. Pittenger AO: Inequalities between arithmetic and logarithmic means. Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta 1980, (678–715):15–18.
16. Imoru CO: The power mean and the logarithmic mean. International Journal of Mathematics and Mathematical Sciences 1982, 5(2):337–343. 10.1155/S0161171282000313
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.2008.03.071
18. Li X, Chen C-P, Qi F: Monotonicity result for generalized logarithmic means. Tamkang Journal of Mathematics 2007, 38(2):177–181.
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.