Research Article | Open Access

# A Sharp Double Inequality between Harmonic and Identric Means

**Academic Editor:**OndΕej DoΕ‘lΓ½

#### Abstract

We find the greatest value and the least value in such that the double inequality holds for all with . Here, , and denote the harmonic and identric means of two positive numbers and , respectively.

#### 1. Introduction

The classical harmonic mean and identric mean of two positive numbers and are defined by respectively. Recently, both mean values have been the subject of intensive research. In particular, many remarkable inequalities for and can be found in the literature [1β17].

Let , , , , and be the th power, logarithmic, geometric, arithmetic, and Seiffert means of two positive numbers and with , respectively. Then it is well-known that for all with .

Long and Chu [18] answered the question: what are the greatest value and the least value such that for all with and with .

In [19], the authors proved that the double inequality holds for all with if and only if and .

The following sharp bounds for , and in terms of power means are presented in [20]:

for all with .

Alzer and Qiu [21] proved that the inequalities hold for all positive real numbers and with if and only if and , and so forth.

For fixed with and , let

Then it is not difficult to verify that is continuous and strictly increasing in . Note that and . Therefore, it is natural to ask what are the greatest value and the least value in such that the double inequality holds for all with . The main purpose of this paper is to answer these questions. Our main result is Theorem 1.1.

Theorem 1.1. *If , then the double inequality
**
holds for all with if and only if and .*

#### 2. Proof of Theorem 1.1

*Proof of Theorem 1.1. *Let and . Then from the monotonicity of the function in we know that to prove inequality (1.8) we only need to prove that inequalities
hold for all with .

Without loss of generality, we assume that . Let and , then from (1.1) and (1.2) one has

Let
Then simple computations lead to
where
where

We divide the proof into two cases. *Case 1 (). *Then (2.19), (2.22), (2.25), and (2.28) lead to

From (2.27) we clearly see that is strictly increasing in , then inequality (2.32) leads to the conclusion that for , hence is strictly increasing in .

It follows from inequality (2.31) and the monotonicity of that is strictly increasing in . Then (2.30) implies that for , so is strictly increasing in .

From (2.29) and the monotonicity of we clearly see that is strictly increasing in .

From (2.5), (2.7), (2.9), (2.11), (2.13), (2.16), and the monotonicity of we conclude that
for .

Therefore, inequality (2.1) follows from (2.3) and (2.4) together with inequality (2.33).*Case 2 (). *Then (2.19), (2.22), (2.25), and (2.28) lead to
From (2.27) and (2.37) we know that is strictly increasing in . Then (2.26) and (2.36) lead to the conclusion that there exists such that for and for , hence is strictly decreasing in and strictly increasing in .

It follows from (2.23) and (2.35) together with the piecewise monotonicity of that there exists such that is strictly decreasing in and strictly increasing in . Then (2.20) and (2.34) lead to the conclusion that there exists such that is strictly decreasing in and strictly increasing in .

From (2.16) and (2.17) together with the piecewise monotonicity of we clearly see that there exists such that for and for . Therefore, is strictly decreasing in and strictly increasing in . Then (2.11)β(2.14) lead to the conclusion that there exists such that is strictly decreasing in and strictly increasing in .

It follows from (2.7)β(2.10) and the piecewise monotonicity of that there exists such that is strictly decreasing in and strictly increasingin .

Note that (2.6) becomes
for .

From (2.5) and (2.38) together with the piecewise monotonicity of we clearly see that
for .

Therefore, inequality (2.2) follows from (2.3) and (2.4) together with inequality (2.39).

Next, we prove that the parameter is the best possible parameter in such that inequality (2.1) holds for all with . In fact, if , then (2.19) leads to . From the continuity of we know that there exists such that
for .

It follows from (2.3)β(2.5), (2.7), (2.9), (2.11), (2.13), and (2.16) that for .

Finally, we prove that the parameter is the best possible parameter in such that inequality (2.2) holds for all with . In fact, if , then (2.6) leads to . Hence, there exists such that
for .

Therefore, for , follows from (2.3) and (2.4) together with inequality (2.41).

#### Acknowledgments

This research was supported by the Natural Science Foundation of China under Grant 11071069 and Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

#### References

- Y. M. Chu and B. Y. Long, βSharp inequalities between means,β
*Mathematical Inequalities and Applications*, vol. 14, no. 3, pp. 647β655, 2011. View at: Google Scholar - Y.-M. Chu, S.-S. Wang, and C. Zong, βOptimal lower power mean bound for the convex combination of harmonic and logarithmic means,β
*Abstract and Applied Analysis*, vol. 2011, Article ID 520648, 9 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M.-K. Wang, Y.-M. Chu, and Y.-F. Qiu, βSome comparison inequalities for generalized Muirhead and identric means,β
*Journal of Inequalities and Applications*, vol. 2010, Article ID 295620, 10 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y.-M. Chu and B.-Y. Long, βBest possible inequalities between generalized logarithmic mean and classical means,β
*Abstract and Applied Analysis*, vol. 2010, Article ID 303286, 13 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y.-M. Chu and W.-F. Xia, βInequalities for generalized logarithmic means,β
*Journal of Inequalities and Applications*, vol. 2009, Article ID 763252, 7 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Alzer, βA harmonic mean inequality for the gamma function,β
*Journal of Computational and Applied Mathematics*, vol. 87, no. 2, pp. 195β198, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Alzer, βAn inequality for arithmetic and harmonic means,β
*Aequationes Mathematicae*, vol. 46, no. 3, pp. 257β263, 1993. View at: Google Scholar | Zentralblatt MATH - H. Alzer, βInequalities for arithmetic, geometric and harmonic means,β
*The Bulletin of the London Mathematical Society*, vol. 22, no. 4, pp. 362β366, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Alzer, βUngleichungen fΓΌr Mittelwerte,β
*Archiv der Mathematik*, vol. 47, no. 5, pp. 422β426, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. SΓ‘ndor, βTwo inequalities for means,β
*International Journal of Mathematics and Mathematical Sciences*, vol. 18, no. 3, pp. 621β623, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. SΓ‘ndor, βOn refinements of certain inequalities for means,β
*Archivum Mathematicum*, vol. 31, no. 4, pp. 279β282, 1995. View at: Google Scholar | Zentralblatt MATH - J. SΓ‘ndor, βOn certain identities for means,β
*Studia Universitatis BabeΕ-Bolyai, Mathematica*, vol. 38, no. 4, pp. 7β14, 1993. View at: Google Scholar | Zentralblatt MATH - J. SΓ‘ndor, βA note on some inequalities for means,β
*Archiv der Mathematik*, vol. 56, no. 5, pp. 471β473, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. SΓ‘ndor, βOn the identric and logarithmic means,β
*Aequationes Mathematicae*, vol. 40, no. 2-3, pp. 261β270, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. R. Mercer, βRefined arithmetic, geometric and harmonic mean inequalities,β
*The Rocky Mountain Journal of Mathematics*, vol. 33, no. 4, pp. 1459β1464, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. K. Vamanamurthy and M. Vuorinen, βInequalities for means,β
*Journal of Mathematical Analysis and Applications*, vol. 183, no. 1, pp. 155β166, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. Gautschi, βA harmonic mean inequality for the gamma function,β
*SIAM Journal on Mathematical Analysis*, vol. 5, pp. 278β281, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B.-Y. Long and Y.-M. Chu, βOptimal power mean bounds for the weighted geometric mean of classical means,β
*Journal of Inequalities and Applications*, vol. 2010, Article ID 905679, 6 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y.-M. Chu, Y.-F. Qiu, M.-K. Wang, and G.-D. Wang, βThe optimal convex combination bounds of arithmetic and harmonic means for the Seiffert's mean,β
*Journal of Inequalities and Applications*, vol. 2010, Article ID 436457, 7 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Alzer, βBestmΓΆgliche AbschΓ€tzungen fΓΌr spezielle Mittelwerte,β
*Zbornik Radova Prirodno-Matematichkog Fakulteta, Serija za Matematiku*, vol. 23, no. 1, pp. 331β346, 1993. View at: Google Scholar | Zentralblatt MATH - H. Alzer and S.-L. Qiu, βInequalities for means in two variables,β
*Archiv der Mathematik*, vol. 80, no. 2, pp. 201β215, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright
Β© 2011 Yu-Ming Chu et al. This is an open access
article distributed under the