Research Article | Open Access
Yu-Ming Chu, Miao-Kun Wang, Song-Liang Qiu, Ye-Fang Qiu, "Sharp Generalized Seiffert Mean Bounds for Toader Mean", Abstract and Applied Analysis, vol. 2011, Article ID 605259, 8 pages, 2011. https://doi.org/10.1155/2011/605259
Sharp Generalized Seiffert Mean Bounds for Toader Mean
For , the generalized Seiffert mean of two positive numbers and is defined by . In this paper, we find the greatest value and least value such that the double inequality holds for all with , and give new bounds for the complete elliptic integrals of the second kind. Here, denotes the Toader mean of two positive numbers and .
For , the generalized Seiffert mean of two positive numbers and is defined by
It is well known that is continuous and strictly increasing with respect to for fixed with . In particular, if , then the generalized Seiffert mean reduces to the Seiffert mean
In , Toader introduced the Toader mean of two positive numbers and as follows: where , is the complete elliptic integral of the second kind.
In , Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows: for all with .
The main purpose of this paper is to find the greatest value and least value such that the double inequality holds for all with and give new bounds for the complete elliptic integrals of the second kind.
In order to establish our main result, we need several formulas and lemmas, which we present in this section.
The following formulas were presented in [10, Appendix E, pages 474-475]: Let , then
Lemma 2.1 (see [10, Theorem 1.25]). For , let be continuous on and be differentiable on , let for all . If is increasing (decreasing) on , then so are If is strictly monotone, then the monotonicity in the conclusion is also strict.
(1) is strictly increasing from onto ;
(2) is strictly increasing from onto ;
(3) is strictly increasing from onto ;
(4) is strictly increasing from onto ;
(5) is strictly increasing from onto ;
(6) is strictly increasing from onto .
Proof. Parts (1)–(4) can be found in [10, Theorem , Theorem , and Exercise and ].
For part (5), clearly (. Let and , then , and
It follows from (2.3) and part (1) together with Lemma 2.1 that is strictly increasing in and .
For part (6), clearly . Let and , then , , and
From (2.4), parts (2) and (5) together with Lemma 2.1, we know that is strictly increasing in , and .
(1) is strictly increasing from onto .
(2) for .
Proof. For part (1), clearly and . Simple computation leads to
Making use of Lemma 2.2 (1), (2), and (6), we get
Therefore, part (1) follows from (2.5) and (2.6) together with the limiting values of at and .
For part (2), simple computations yield that where . Note that where .
From Lemma 2.2(1), (3), and (4) together with the monotonicity of we know that is strictly increasing in . Moreover,
Equations (2.11)–(2.13) and the monotonicity of lead to the conclusion that there exists such that is strictly decreasing in and strictly increasing in .
It follows from (2.8)–(2.10) and the piecewise monotonicity of that there exists such that is strictly decreasing in and strictly increasing in .
Therefore, part follows from (2.7) and the piecewise monotonicity of .
3. Main Result
Theorem 3.1. Inequality holds for all with , and and are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean , respectively.
Proof. Firstly, we prove that
for all with .
Without loss of generality, we assume that . Let , . Then (1.1) and (1.3) lead to where and are defined as in Lemma 2.3.
Therefore, inequality (3.1) follows from (3.2) and (3.3) together with Lemma 2.3.
Next, we prove that and are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean , respectively.
For any and , from (1.1) and (1.3) one has where .
Letting and making use of Taylor expansion, we get
Inequality (3.4) and equations (3.5) and (3.6) imply that for any there exist and , such that for and for .
From Theorem 3.1, we get new bounds for the complete elliptic integrals of the second kind as follows.
Corollary 3.2. The inequality holds for all .
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. The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions.
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