Research Article | Open Access
Wei-Dong Jiang, "Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean", The Scientific World Journal, vol. 2013, Article ID 842542, 4 pages, 2013. https://doi.org/10.1155/2013/842542
Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean
Abstract
The authors find the greatest value and the least value , such that the double inequality holds for all and with , where , , and denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers and .
1. Introduction
In [1], Toader introduced a mean where for is the complete elliptic integral of the second kind.
In recent years, there have been plenty of literature, such as [2–6], dedicated to the Toader mean.
For and , the centroidal mean and th power mean are, respectively, defined by
In [7], Vuorinen conjectured that for all with . This conjecture was verified by Qiu and Shen [8] and by Barnard et al. [9], respectively.
In [10], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: for all with .
Chu et al. [5] proved that the double inequality holds for all with if and only if and .
Very recently, Hua and Qi [11] proved that the double inequality is valid for all with if and only if and . Where denote the arithmetic mean.
For positive numbers with , let be on . It is not difficult to directly verify that is continuous and strictly increasing on .
The main purpose of the paper is to find the greatest value and the least value , such that the double inequality holds for all and with . As applications, we also present new bounds for the complete elliptic integral of the second kind.
2. Preliminaries and Lemmas
In order to establish our main result, we need several formulas and Lemmas below.
For and , Legendre’s complete elliptic integrals of the first and second kinds are defined in [12, 13] by respectively.
For , the formulas were presented in [14, Appendix E, pages 474-475].
Lemma 1 (see [14, Theorem 3.21(1), 3.43 exercises 13(a)]). The function   is strictly increasing from to , and the function is increasing from to .
Lemma 2. Let and Then, , for all if and only if , and , for all if and only if .
Proof. From (11), one has
where .
We divide the proof into four cases.
Case 1 ().  From (14) and Lemma 1 together with the monotonicity of , we clearly see that is strictly increasing on . Therefore, , for all .
Case 2 (). From (14) and Lemma 1 together with the monotonicity of , we obtain that is strictly decreasing on . Therefore, , for all .
Case 3 (). From (13) and (14) together with the monotonicity of , we see that there exists , such that is strictly increasing in and strictly decreasing in and
Therefore, making use of (12) and inequality (15) together with the piecewise monotonicity of leads to the conclusion that there exists , such that for and for .
Case 4 (). Equation (13) leads to
From (13) and (14) together with the monotonicity of , we clearly see that there exists , such that is strictly increasing in and strictly decreasing in . Therefore, for follows from (12) and (16) together with the piecewise monotonicity of .
3. Main Results
Now, we are in a position to state and prove our main results.
Theorem 3. If and , then the double inequality holds for all with if and only if
Proof. Since , , and are symmetric and homogeneous of degree one, without loss of generality, we assume that . Let , , and . Then, Therefore, Theorem 3 follows easily from Lemma 2 and (19).
Let ,  ,  . Then, from Theorem 3, we get new bounds for the complete elliptic integral of the second kind in terms of elementary functions as follows.
Corollary 4. For and , one has
4. Remarks
Remark 5. In the recent past, the complete elliptic integrals have attracted the attention of numerous mathematicians. In [4], it was established that
for all .
Guo and Qi [15] proved that
for all .
Yin and Qi [16] presented that
for all .
It was pointed out in [4] that the bounds in (21) for are better than the bounds in (22) for some .
Remark 6. The lower bound in (20) for is better than the lower bound in (21). Indeed, for all .
Remark 7. The following equivalence relations for show that the lower bound in (20) for is better than the lower bound in (23):
Acknowledgments
The author is thankful to the anonymous referees for their valuable and profound comments on and suggestions to the original version of this paper. This work was supported by the project of Shandong Higher Education Science and Technology Program under Grant no. J11LA57.
References
- G. Toader, “Some mean values related to the arithmetic-geometric mean,” Journal of Mathematical Analysis and Applications, vol. 218, no. 2, pp. 358–368, 1998. View at: Publisher Site | Google Scholar
- Y.-M. Chu and M.-K. Wang, “Inequalities between arithmetic-geometric, Gini, and Toader means,” Abstract and Applied Analysis, vol. 2012, Article ID 830585, 11 pages, 2012. View at: Publisher Site | Google Scholar
- Y.-M. Chu and M.-K. Wang, “Optimal lehmer mean bounds for the Toader mean,” Results in Mathematics, vol. 61, no. 3-4, pp. 223–229, 2012. View at: Publisher Site | Google Scholar
- Y.-M. Chu, M.-K. Wang, and S.-L. Qiu, “Optimal combinations bounds of root-square and arithmetic means for Toader mean,” Proceedings of the Indian Academy of Sciences, vol. 122, no. 1, pp. 41–51, 2012. View at: Publisher Site | Google Scholar
- Y.-M. Chu, M.-K. Wang, and X.-Y. Ma, “Sharp bounds for Toader mean in terms of contraharmonic mean with applications,” Journal of Mathematical Inequalities, vol. 7, no. 2, pp. 161–166, 2013. View at: Google Scholar
- Y.-M. Chu, M.-K. Wang, S.-L. Qiu, and Y.-F. Qiu, “Sharp generalized seiffert mean bounds for toader mean,” Abstract and Applied Analysis, vol. 2011, Article ID 605259, 8 pages, 2011. View at: Publisher Site | Google Scholar
- M. Vuorinen, “Hypergeometric functions in geometric function theory,” in Proceedings of the Workshop on Special Functions and Differential Equations, pp. 119–126, The Institute of Mathematical Sciences, Madras, India, January 1998. View at: Google Scholar
- S.-L. Qiu and J.-M. Shen, “On two problems concerning means,” Journal of Hangzhou Insitute of Electronic Engineering, vol. 17, no. 3, pp. 1–7, 1997 (Chinese). View at: Google Scholar
- R. W. Barnard, K. Pearce, and K. C. Richards, “An inequality involving the generalized hypergeometric function and the arc length of an ellipse,” SIAM Journal on Mathematical Analysis, vol. 31, no. 3, pp. 693–699, 2000. View at: Google Scholar
- H. Alzer and S.-L. Qiu, “Monotonicity theorems and inequalities for the complete elliptic integrals,” Journal of Computational and Applied Mathematics, vol. 172, no. 2, pp. 289–312, 2004. View at: Publisher Site | Google Scholar
- Y. Hua and F. Qi, “The best bounds for toader mean in terms of the centroidal and arithmetic mean,” http://arxiv.org/abs/1303.2451. View at: Google Scholar
- F. Bowman, Introduction to Elliptic Functions with Applications, Dover, New York, NY, USA, 1961.
- P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, New York, NY, USA, 1971.
- G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, NY, USA, 1997.
- B.-N. Guo and F. Qi, “Some bounds for the complete elliptic integrals of the first and second kinds,” Mathematical Inequalities and Applications, vol. 14, no. 2, pp. 323–334, 2011. View at: Google Scholar
- L. Yin and F. Qi, “Some inequalities for complete elliptic integrals,” http://arxiv.org/abs/1301.4385. View at: Google Scholar
Copyright
Copyright © 2013 Wei-Dong Jiang. 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.