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Iulia Costin, Gheorghe Toader, "A Nice Separation of Some Seiffert-Type Means by Power Means", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 430692, 6 pages, 2012. https://doi.org/10.1155/2012/430692
A Nice Separation of Some Seiffert-Type Means by Power Means
Seiffert has defined two well-known trigonometric means denoted by and . In a similar way it was defined by Carlson the logarithmic mean as a hyperbolic mean. Neuman and Sándor completed the list of such means by another hyperbolic mean . There are more known inequalities between the means , and and some power means . We add to these inequalities two new results obtaining the following nice chain of inequalities , where the power means are evenly spaced with respect to their order.
A mean is a function , with the property Each mean is reflexive; that is, This is also used as the definition of .
We will refer here to the following means:(i)the power means , defined by (ii)the geometric mean , defined as , but verifying also the property (iii)the first Seiffert mean , defined in  by (iv)the second Seiffert mean , defined in  by (v)the Neuman-Sándor mean , defined in  by (vi)the Stolarsky means defined in  as follows:
The mean is the arithmetic mean and the mean is the logarithmic mean. As Carlson remarked in , the logarithmic mean can be represented also by thus the means , and are very similar. In  it is also proven that these means can be defined using the nonsymmetric Schwab-Borchardt mean given by (see [6, 7]). It has been established in  that
2. Interlacing Property of Power Means
Given two means and , we will write if
It is known that the family of power means is an increasing family of means, thus Of course, it is more difficult to compare two Stolarsky means, each depending on two parameters. To present the comparison theorem given in [8, 9], we have to give the definitions of the following two auxiliary functions:
Theorem 2.1. Let . Then the comparison inequality holds true if and only if , and (1) if , (2) if , or (3) if .
We need also in what follows an important double-sided inequality proved in  for the Schwab-Borchardt mean:
Being rather complicated, the Seiffert-type means were evaluated by simpler means, first of all by power means. The evaluation of a given mean by power means assumes the determination of some real indices and such that . The evaluation is optimal if is the the greatest and is the smallest index with this property. This means that cannot be compared with if .
For the logarithmic mean in , it was determined the optimal evaluation For the Seiffert means, there are known the evaluations proved in  and given in . It is also known that as it was shown in . Moreover in  it was determined the optimal evaluation Using these results we deduce the following chain of inequalities: To prove the full interlacing property of power means, our aim is to show that can be put between and . We thus obtain a nice separation of these Seiffert-type means by power means which are evenly spaced with respect to their order.
3. Main Results
We add to the inequalities (2.11) the next results.
Theorem 3.1. The following inequalities are satisfied.
Proof. First of all, let us remark that . So, for the first inequality in (3.1), it is sufficient to prove that the following chain of inequalities is valid. The first inequality in (3.2) is a simple consequence of the property of the mean given in (1.11) and the second inequality from (2.5). The second inequality can be proved by direct computation or by taking which gives which is easy to prove. The last inequality in (3.2) is given by the comparison theorem of the Stolarsky means. In a similar way, the second inequality in (3.1) is given by the relations The first inequality is again given by the comparison theorem of the Stolarsky means. The equality in (3.4) is shown by elementary computations, and the last inequality is a simple consequence of the property of the mean given in (1.11) and the first inequality from (2.5).
Corollary 3.2. The following two-sided inequality is valid for all .
The authors wish to thank the anonymous referee for offering them a simpler proof for their results.
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Copyright © 2012 Iulia Costin and Gheorghe Toader. 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.