`Abstract and Applied AnalysisVolumeΒ 2011, Article IDΒ 679201, 9 pageshttp://dx.doi.org/10.1155/2011/679201`
Research Article

Sharp Bounds for Power Mean in Terms of Generalized Heronian Mean

College of Mathematics and Computer Science, Hebei University, Baoding 071002, China

Received 2 March 2011; Accepted 7 April 2011

Copyright Β© 2011 Hongya Gao 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.

Abstract

For , we find the least value and the greatest value such that the inequality holds for all with . Here, and are the generalized Heronian and the power means of two positive numbers and , respectively.

1. Introduction and Statement of Result

For with , the generalized Heronian mean of and is defined by Janous [1] as If we take in (1.1), then we arrive at the classical Heronian mean The domain of definition for the function can be extended to all with , that is, For all fixed , it is easy to derive that , is monotonically decreasing, and

Let denote the power mean of order . In particular, the harmonic, geometric, square-root, arithmetic, and root-square means of and are It is well known that the power mean of order given in (1.5) is monotonically increasing in , then we can write

Recently, the inequalities for means have been the subject of intensive research [1β15]. In particular, many remarkable inequalities for the generalized Heronian and power means can be found in the literature [4β9].

In [4], the authors established two sharp inequalities

In [5], Long and Chu found the greatest value and the least value such that the double inequality holds for all and with .

In [6], Shi et al. gave two optimal inequalities for , where is the logarithmic mean for .

In [7], Guan and Zhu obtained sharp bounds for the generalized Heronian mean in terms of the power mean with . The optimal values and such that holds in general are (1)in case of and ,(2) in case of and .

In this paper, we find the least value and the greatest value , such that for any fixed , the inequality holds for all with .

Theorem 1.1. For , the optimal numbers and such that is valid for all with , are and .

Notice that in our case ; the two numbers and are all negative see Corollary 2.2 below. Thus, the result in this paper is different from [7, Theorem A].

2. Preliminary Lemmas

The following lemma will be repeatedly used in the proof of Theorem 1.1.

Lemma 2.1. For , one has

Proof. We show that which is clearly equivalent to the claim. Equation (2.2) follows from the facts

Corollary 2.2. If , then

Proof. Since for , the two functions are strictly decreasing, then one has It suffices to show that which is equivalent to (2.1).

Lemma 2.3. For and , let Then, is strictly decreasing for , and

Proof. The fact for and is obvious, which allows us to take the logarithmic function of , Some tedious, but not difficult calculations lead to where It is easy to see that Equation (2.14) implies that is strictly decreasing for , which together with (2.13) implies for . Thus, by (2.11), which implies Hence, is strictly decreasing.
It remains to show (2.9). The first equality in (2.9) is obvious. The second one follows from This ends the proof of Lemma 2.3.

Lemma 2.4. For , , and , let Then,

Proof. Simple calculations lead to where we have used L'Hospital's law. This ends the proof of Lemma 2.4.

3. Proof of Theorem 1.1

Proof. Firstly, we prove that for , hold true for all with . It is no loss of generality to assume that . Let and . In view of Corollary 2.2, . Equations (1.3) and (1.5) lead to where is defined by (2.18). It is easy to see that By Lemma 2.3,
We now distinguish between two cases.
Case 1 (). Since , then by (3.7), . Thus, is strictly increasing for , which together with (3.6) implies . Hence, is strictly increasing for . Since (3.4), then . Equation (3.1) follows from (3.3).Case 2 (). By (3.5) and (2.11), Thus, is strictly increasing. Equations (3.8) and (2.1) imply Equations (3.7) and (2.9) imply Combining (3.10) with (3.11), we obviously know that there exists such that for and for . This implies that is strictly decreasing for and strictly increasing for . By (3.6) and Lemma 2.4, we know that for . Therefore, is strictly decreasing. By (3.4) and Lemma 2.4 again, we derive that for . Equation (3.2) follows from (3.3).Secondly, we prove that is the best lower bound for the power mean for . For any , Hence, there exists such that for .
Finally, we prove that is the best upper bound for the power mean for . For any , by (3.7) (with in place of ), we have Hence, by the continuity of , there exists such that for . Thus is strictly decreasing for . From (3.6), for . This result together with (3.4) implies that for . Hence, by (3.3), for .

Acknowledgments

This work was supported by NSFC (10971224) and NSF of Hebei Province (A2011201011).

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