A Nice Separation of Some Seiffert-Type Means by Power Means

Seiffert has defined two well-known trigonometric means denoted by P and T. In a similar way it was defined by Carlson the logarithmic mean L as a hyperbolic mean. Neuman and Sándor completed the list of such means by another hyperbolic mean M. There are more known inequalities between the means P, T, and L and some power means Ap. We add to these inequalities two new results obtaining the following nice chain of inequalities A0 < L < A1/2 < P < A1 < M < A3/2 < T < A2, where the power means are evenly spaced with respect to their order.


Means
A mean is a function M : R 2 → R , with the property min a, b ≤ M a, b ≤ max a, b , ∀a, b > 0. 1.1 Each mean is reflexive; that is, M a, a a, ∀a > 0.
1.2 This is also used as the definition of M a, a .We will refer here to the following means: vi the Stolarsky means S p,q defined in 4 as follows: S p,q a, b

1.8
The mean A 1 A is the arithmetic mean and the mean S 1,0 L is the logarithmic mean.As Carlson remarked in 5 , the logarithmic mean can be represented also by L a, b a − b thus the means P, T, M, and L are very similar.In 3 it is also proven that these means can be defined using the nonsymmetric Schwab-Borchardt mean SB given by

Interlacing Property of Power Means
Given two means M and N, we will write 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:

2.3
Theorem 2.1.Let p, q, r, s ∈ R. Then the comparison inequality holds true if and only if p q ≤ r s, and (1) l p, q ≤ l r, s if 0 ≤ min p, q, r, s , (2) k p, q ≤ k r, s if min p, q, r, s < 0 < max p, q, r, s , or (3) −l −p, −q ≤ −l −r, −s if max p, q, r, s ≤ 0.
We need also in what follows an important double-sided inequality proved in 3 for the Schwab-Borchardt mean:

International Journal of Mathematics and Mathematical Sciences
Being rather complicated, the Seiffert-type means were evaluated by simpler means, first of all by power means.The evaluation of a given mean M by power means assumes the determination of some real indices p and q such that A p < M < A q .The evaluation is optimal if p is the the greatest and q is the smallest index with this property.This means that M cannot be compared with A r if p < r < q.
For the logarithmic mean in 10 , it was determined the optimal evaluation For the Seiffert means, there are known the evaluations proved in 11 and given in 2 .It is also known that as it was shown in 3 .Moreover in 12 it was determined the optimal evaluation A ln 2/ ln π < P < A 2/3 .

2.10
Using these results we deduce the following chain of inequalities:

2.11
To prove the full interlacing property of power means, our aim is to show that A 3/2 can be put between M and T. We thus obtain a nice separation of these Seiffert-type means by power means which are evenly spaced with respect to their order.

Main Results
We add to the inequalities 2.11 the next results.Proof.First of all, let us remark that A 3/2 S 3,3/2 .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 M given in 1.11 and the second inequality from 2.5 .The second inequality can be proved by direct computation or by taking a 1 t, b 1 − t, 0 < t < 1 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 S 3,3/2 < S 4,1 3 AA 2 2 < T.

3.4
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 T given in 1.11 and the first inequality from 2.5 .

i 3 2International 6 v
the power means A p , defined by A p a, b a p b p 2 1/p , p / 0; 1.Journal of Mathematics and Mathematical Sciences ii the geometric mean G, defined as G a, b √ ab, but verifying also the property lim p → 0 A p a, b A 0 a, b G a, b ; 1.4 iii the first Seiffert mean P, defined in 1 by P a, b a − b 2 sin −1 a − b / a b , a / b; 1.5 iv the second Seiffert mean T, defined in 2 by T a, b a − b 2 tan −1 a − b / a b , a / b; 1.the Neuman-Sándor mean M, defined in 3 by M a, b a − b 2 sinh −1 a − b / a b , a / b; 1.7