Optimal bounds for the Neuman-Sándor mean in terms of the first Seiffert and quadratic means
© Gong et al.; licensee Springer. 2013
Received: 12 July 2013
Accepted: 18 October 2013
Published: 22 November 2013
In this paper, we find the least value α and the greatest value β such that the double inequality
holds true for all with , where , and are the first Seiffert, Neuman-Sándor and quadratic means of a and b, respectively.
Let u, v and w be the bivariate means such that for all with . The problems of finding the best possible parameters α and β such that the inequalities and hold for all with have attracted the interest of many mathematicians.
respectively. In here, is the inverse hyperbolic sine function.
hold for all with .
hold for all and with and .
Li et al.  proved that the double inequality holds for all with , where (), and is the p th generalized logarithmic mean of a and b, and is the unique solution of the equation .
hold for all with if , , and .
hold for all with and , where .
Inspired by inequalities (1.3) and (1.4), in this paper, we present the optimal upper and lower bounds for the Neuman-Sándor mean in terms of the geometric convex combinations of the first Seiffert mean and the quadratic mean . All numerical computations are carried out using Mathematica software.
In order to establish our main result, we need several lemmas, which we present in this section.
holds for .
Therefore, inequality (2.2) follows from (2.4)-(2.7), and inequality (2.3) follows from (2.4)-(2.6) and (2.8). □
holds for .
Therefore, Lemma 2.2 follows from (2.9). □
holds for , where is the inverse sine function.
Therefore, inequality (2.10) follows from (2.12), (2.14), (2.15) and (2.19), and inequality (2.11) follows from (2.12) and (2.14)-(2.16) together with (2.18). □
holds for .
Therefore, inequality (2.22) follows from (2.24), (2.25) and (2.29), and inequality (2.23) follows from (2.24), (2.25) and (2.30). □
holds for .
for and for .
Therefore, inequality (2.31) follows from (2.33), (2.35), (2.36) and (2.40), and inequality (2.32) follows from (2.33)-(2.36) and (2.41). □
Then for .
for . □
Then for .
for follows from (2.53) and (2.54).
for . □
Lemma 2.8 Let , and , where and are defined as in Lemmas 2.4 and 2.5, respectively. Then the function is strictly decreasing on .
for . This in turn implies that is strictly decreasing on . □
3 Main result
holds for all with if and only if and .
where and are defined as in Lemmas 2.4 and 2.5, respectively.
Therefore, Theorem 3.1 follows from (3.13) and (3.14) together with the following statements:
If , then (3.1) and (3.2) imply that there exists such that for all with .
If , then (3.1) and (3.3) imply that there exists such that for all with .
This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, and the Natural Science Foundation of Zhejiang Province under Grants LY13H070004 and LY13A010004.
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