Research Article | Open Access
Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean
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 .
In , Toader introduced a mean where for is the complete elliptic integral of the second kind.
For and , the centroidal mean and th power mean are, respectively, defined by
In , Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: for all with .
Chu et al.  proved that the double inequality holds for all with if and only if and .
Very recently, Hua and Qi  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 , 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
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
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
Remark 5. In the recent past, the complete elliptic integrals have attracted the attention of numerous mathematicians. In , it was established that
for all .
Guo and Qi  proved that for all .
Yin and Qi  presented that for all .
It was pointed out in  that the bounds in (21) for are better than the bounds in (22) for some .
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.
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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.