Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 489490, 6 pages
http://dx.doi.org/10.1155/2015/489490

Research Article

Optimal Bounds for the Neuman-Sándor Mean in terms of the Convex Combination of the First and Second Seiffert Means

Hao-Chuan Cui, Nan Wang, and Bo-Yong Long

School of Mathematical Science, Anhui University, Hefei 230601, China

Received 15 June 2015; Accepted 21 July 2015

Academic Editor: Dapeng P. Du

Copyright © 2015 Hao-Chuan Cui 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.

We find the least value α and the greatest value β such that the double inequality holds for all with , where , and are the Neuman-Sándor mean and the first and second Seiffert means of two positive numbers and , respectively.

1. Introduction

For with , the Neuman-Sándor mean [1], the first Seiffert mean [2], and the second Seiffert mean [3] are defined by

respectively. It can be observed that the first Seiffert mean can be rewritten as (see [1])

where ), , and are the inverse hyperbolic sine, inverse tangent, and inverse sine functions, respectively.

Recently, the means , , and and other means have been the subject of intensive research. Many remarkable inequalities for means can be found in the literature [4–14].

Let , , , , , , and

denote the harmonic, geometric, logarithmic, identric, arithmetic, root-square, and th power means of two positive numbers and with , respectively. Then it is well known that the inequalities

hold for with .

Neuman and Sándor [1] established

for all with and the following Ky Fan inequalities:

for all with , and .

Gao [15] proved the optimal double inequalities

hold for all with .

The following bounds for the Seiffert means and in terms of power mean were presented by Jagers in [16]:

for all with . Hästö [17] improved the results of [16] and found the sharp lower power mean bound for the Seiffert mean as follows:

for all with .

In [18], the authors proved the sharp double inequality

holds for all with .

Let denote the Lehmer mean of two positive numbers and with . In [19] the authors presented the following best possible Lehmer mean bounds for the Seiffert means and :

for all with .

Let , , and be bivariate means such that for all with . The problems of finding the best possible parameters and such that the inequalities and hold for all with have attracted the interest of many mathematicians.

In [20, 21] the authors proved that the double inequalities

hold for all with if and only if , , , and .

In [22], the authors proved that the double inequality

holds for all with if and only if , .

The main purpose of this paper is to find the least value and the greatest value such that the double inequality

holds for all with .

2. Lemmas

To establish our main result, we need several lemmas which we present in this section.

Lemmas 1 and 2 were proved in [23].

Lemma 1. Let . Then is strictly decreasing on . Moreover, the inequality

holds for any and the inequality

holds for any .

Lemma 2. Let . Then it holds that

for any and

holds for any .

For , the power series expansion of the function is presented as follows:

Therefore, we have the following.

Lemma 3. If , then it holds that

Proof. The inequality of (22) follows from equality (21) directly. The inequality of (23) follows from

for .

Lemma 4. Let . Then for it holds that

Proof. To prove inequality (25), it suffices to show

for , where

Firstly, we prove inequality (26). From the expression of , we have

where

Because of

for , it follows from (22) and (31) that

for any . Therefore, inequality (26) follows from (29) and (30) together with (33).
Secondly, we prove inequality (27). From the expression of , we have

where

Because of

for any , it follows from (22) and (35) together with (36) that

for any . Therefore, inequality (27) follows from (34) and (37).

Lemma 5. The function is strictly decreasing on , where and , , and are defined as in Lemmas 1, 2, and 4, respectively.

Proof. Direct computations lead to

where

Then inequality (22) leads to

for any Obviously, it holds that

for any . Therefore, for any , follows from (39) and (41) together with (42). In other words, is strictly decreasing on ().
Let . It was proved that is strictly decreasing on () in [23] Lemma . Thus, from the monotonicity of and , we have

for any . That is to say, is strictly decreasing on (). Considering the monotonicity of in Lemma 1, the proof is completed.

Lemma 6. Let , where . Then for .

Proof. Let . Then it is easy to verify that is decreasing on , where

Considering , we have for . Therefore, for .

3. Main Results

Theorem 7. The double inequality

holds for any 0 with if and only if and .

Proof. Because , , and are symmetric and homogeneous of degree 1, without loss of generality, we assume that . Let , , and . Then by (1), (3), and (4), direct computations lead to

Let

Then it follows that

Differentiating , we have

where , , and are defined as in Lemmas 1, 2, and 4, respectively.
On one hand, from inequalities in Lemmas 1, 2, and 4, we clearly see that

for . It leads to

for any . According to (47) and (51), we conclude that

for all with .
On the other hand, from inequalities in Lemmas 1, 2, and 4, we have

for . According to Lemma 6, we have

for . Lemma 5 shows that is strictly decreasing on . This fact and (54) together with imply that there exists such that is strictly increasing on and strictly decreasing on . Equations (47) and (48) together with the piecewise monotonicity of lead to the conclusion that

for all with .
At last, by easy computations, (1), (3), and (4) lead to

Thus, we have the following claims.
Claim 1. If , then (56) and (57) imply that there exists such that for all with .
Claim 2. If , then (56) and (58) imply that there exists such that for all with .
Inequalities (52) and (55) in conjunction with the above two claims mean the proof is completed.

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

This research was supported by the Program of Scientific Research Training for Undergraduate of Anhui University (KYXL2014002) and Doctoral Scientific Research Foundation of Anhui University, China (01001901).

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