Research Article | Open Access

Wei-Feng Xia, Yu-Ming Chu, Gen-Di 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. https://doi.org/10.1155/2010/604804

# The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

**Academic Editor:**Lance Littlejohn

#### Abstract

For , the power mean of order , logarithmic mean , and arithmetic mean of two positive real values and are defined by , for and , for , , for and , for and , respectively. In this paper, we answer the question: for , what are the greatest value and the least value , such that the double inequality holds for all ?

#### 1. Introduction

For , the power mean of order and logarithmic mean of two positive real values and are defined by

respectively. In the recent past, both mean values have been the subject of intensive research. In particular, many remarkable inequalities for power mean or logarithmic mean can be found in the literature [1–15]. It might be surprising that the logarithmic mean has applications in physics, economics, and even in meteorology [16–18]. In [16] the authors study a variant of Jensen’s functional equation involving , which appears in a heat conduction problem. A representation of as an infinite product and an iterative algorithm for computing the logarithmic mean as the common limit of two sequences of special geometric and arithmetic means are given in [11]. In [19, 20] it is shown that can be expressed in terms of Gauss's hypergeometric function . And, in [20] the authors prove that the reciprocal of the logarithmic mean is strictly totally positive, that is, every determinant with elements , where and , is positive for all

Let , , and be the arithmetic, geometric, and harmonic means of two positive numbers and , respectively. Then it is well known that

and all inequalities are strict for .

In [21], Alzer and Janous established the following best possible inequality:

for all

In [11, 13, 22] the authors present bounds for in terms of and

for all with .

The following sharp bounds for in terms of power means are proved by Lin [12]

The main purpose of this paper is to answer the question: for , what are the greatest value and the least value , such that the double inequality holds for all

#### 2. Lemmas

In order to establish our results we need several lemmas, which we present in this section.

Lemma 2.1. *If , then *

*Proof. * For , let , then simple computations lead to

From (2.2) we clearly see that for , and for . Then from (2.1) we get

for .

Therefore for follows from (2.3).

Lemma 2.2. *Let , if , then
*

*Proof. * For , let , then and
where

To prove Lemma 2.2 we need only to prove that for . Elementary calculations yield that

for .

Making use of a computer and the mathematica software, from (2.10) we get

From (2.14)–(2.16) we clearly see that there exists a unique , such that for and for . Hence we know that is strictly increasing in and strictly decreasing in .

From (2.11), (2.12), (2.17), (2.18) and the monotonicity of in and in we know that there exist exactly two numbers with , such that for and for , and satisfies

Hence, we know that is strictly decreasing in and strictly increasing in .

Making use of a computer and the mathematica software, from (2.7) and (2.19), we get

Now, (2.8), (2.9), (2.20) and the monotonicity of in and in imply that

for .

Therefore, for follows from (2.6) and (2.21).

Lemma 2.3. * For and , one has the following. *(1)*If then there exists such that for and for .*(2)*If , then for . *

*Proof. * Let , , , , and , then simple computations lead to

If , then from (2.31), (2.34), (2.37)–(2.39), and Lemmas 2.1-2.2 we clearly see that

and is strictly decreasing in .

From (2.43) and the monotonicity of we know that is strictly decreasing in .

The monotonicity of and (2.42) implies that for , then we conclude that is strictly decreasing in .

From the monotonicity of and (2.35) together with (2.41) we clearly see that there exists , such that for and for . Hence we know that is strictly increasing in and strictly decreasing in .

The monotonicity of in and in together with (2.32) and (2.40) imply that there exists , such that for and for . Then we know that is strictly increasing in and strictly decreasing in .

From (2.28) and (2.29) together with the monotonicity of in and in we clearly see that there exists , such that for and for . Hence we know that is strictly increasing in and strictly decreasing in .

Equations (2.25) and (2.26) together with the monotonicity of in and in imply that there exists , such that for and for . Then we conclude that is strictly increasing in and strictly decreasing in .

Now (2.22), (2.23) and the monotonicity of in and in imply that there exists , such that for and for .

(2) If , then (2.31), (2.34), and (2.37)–(2.39) lead to

and is strictly decreasing in .

Therefore, Lemma 2.3(2) follows from (2.22), (2.25), (2.28), (2.44), (2.45), and the monotonicity of .

#### 3. Main Result

Theorem 3.1. *For , the double inequality holds for all , each inequality becomes an equality if and only if , and the given parameters and in each inequality are best possible.*

*Proof. * If , then from (1.1) and (1.2) we clearly see that for . Next, we assume that .

Firstly, we prove that for with .

Without loss of generality, we assume that . Let and , then (1.1) and (1.2) leads to

Let

then
where .

If , then it is not difficult to verify that

From (3.4) and Lemma 2.3(1) we know that there exists , such that is strictly increasing in and strictly decreasing in . Then (3.3) and (3.5) together with the monotonicity of in and in imply that for , and from (3.1) and (3.2) we know that for all with .

If , then from Lemma 2.3(2) and (3.1)–(3.4) we clearly see that for all with .

Secondly, we prove that the parameters and cannot be improved in each inequality.

For any and , from (1.1) and (1.2) we get

Inequality (3.6) implies that for any there exists , such that for . Hence the parameter cannot be improved in the left-side inequality.

Next for , let , then (1.1) and (1.2) leads to

where .

Let , making use of the Taylor expansion we get

Equations (3.7) and (3.8) imply that for any there exists , such that for . Hence the parameter cannot be improved in the right-side inequality.

#### Acknowledgment

This research is supported by the Innovation Team Foundation (no. T200924) and NSF (no. Y200908671) of the Department of Education of Zhejiang Province, and NSF (nos. Y7080106, Y7080185) of Zhejiang Province.

#### References

- 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. View at: Publisher Site | Google Scholar - B. Y. Long and Y. M. Chu, “Optimal power mean bounds for the weighted geometric mean of classical means,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 905679, 8 pages, 2010. View at: Google Scholar - 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. View at: Publisher Site | Google Scholar - Y. M. Chu and W. F. Xia, “Two sharp inequalities for power mean, geometric mean, and harmonic mean,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 741923, 6 pages, 2009. View at: Publisher Site | Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | MathSciNet - S. H. Wu, “Generalization and sharpness of the power means inequality and their applications,”
*Journal of Mathematical Analysis and Applications*, vol. 312, no. 2, pp. 637–652, 2005. 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 | Zentralblatt MATH | MathSciNet - A. O. Pittenger, “Inequalities between arithmetic and logarithmic means,”
*Publikacije Elektrotehničkog Fakulteta. Univerzitet u Beogradu. Serija Matematika I Fizika*, no. 678–715, pp. 15–18, 1980. View at: Google Scholar | Zentralblatt MATH | MathSciNet - P. S. Bullen, D. S. Mitrinović, and P. M. Vasić,
*Means and Their Inequalities*, vol. 31 of*Mathematics and Its Applications (East European Series)*, D. Reidel, Dordrecht, The Netherlands, 1988. View at: MathSciNet - F. Burk, “The Geometric, logarithmic, and arithmetic mean inequality,”
*The American Mathematical Monthly*, vol. 94, no. 6, pp. 527–528, 1987. View at: Publisher Site | Google Scholar | MathSciNet - B. C. Carlson, “The logarithmic mean,”
*The American Mathematical Monthly*, vol. 79, pp. 615–618, 1972. View at: Publisher Site | Google Scholar | Zentralblatt MATH | 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 | Zentralblatt MATH | MathSciNet - J. Sándor, “A note on some inequalities for means,”
*Archiv der Mathematik*, vol. 56, no. 5, pp. 471–473, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. B. Stolarsky, “Generalizations of the logarithmic mean,”
*Mathematics Magazine*, vol. 48, pp. 87–92, 1975. View at: Google Scholar | Zentralblatt MATH | 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 | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. O. Pittenger, “The logarithmic mean in $n$
variables,”
*The American Mathematical Monthly*, vol. 92, no. 2, pp. 99–104, 1985. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Pólya and G. Szegö,
*Isoperimetric Inequalities in Mathematical Physics*, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, NJ, USA, 1951. View at: MathSciNet - B. C. Carlson, “Algorithms involving arithmetic and geometric means,”
*The American Mathematical Monthly*, vol. 78, pp. 496–505, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. C. Carlson and J. L. Gustafson, “Total positivity of mean values and hypergeometric functions,”
*SIAM Journal on Mathematical Analysis*, vol. 14, no. 2, pp. 389–395, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Alzer and W. Janous, “Solution of problem 8,”
*Crux Mathematicorum*, vol. 13, pp. 173–178, 1987. View at: Google Scholar - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2010 Wei-Feng Xia 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.