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

Authors: Iulia Costin; Gheorghe Toader
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

Seiffert has defined two well-known trigonometric means denoted by𝒫 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.


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## Body

## 1.Means

Amean is a function M:ℝ+2→ℝ+, with the property
(1.1)min⁡⁡(a,b)≤M(a,b)≤max⁡⁡(a,b),∀a,b>0.
Each mean is reflexive; that is,
(1.2)M(a,a)=a,∀a>0.
This is also used as the definition of M(a,a).We will refer here to the following means:(i)
the power means𝒜p, defined by
(1.3)Ap(a,b)=[ap+bp2]1/p,p≠0;(ii)
the geometric mean𝒢, defined as 𝒢(a,b)=ab, but verifying also the property
(1.4)lim⁡p→0⁡Ap(a,b)=A0(a,b)=G(a,b);(iii)
the first Seiffert mean𝒫, defined in [1] by
(1.5)P(a,b)=a-b2sin⁡-1⁡((a-b)/(a+b)),a≠b;(iv)
the second Seiffert mean𝒯, defined in [2] by
(1.6)T(a,b)=a-b2tan-1⁡((a-b)/(a+b)),a≠b;(v)
the Neuman-Sándor meanℳ, defined in [3] by
(1.7)M(a,b)=a-b2sinh⁡-1⁡((a-b)/(a+b)),a≠b;(vi)
the Stolarsky means𝒮p,q defined in [4] as follows:
(1.8)Sp,q(a,b)={[q(ap-bp)p(aq-bq)]1/(p-q),pq(p-q)≠01ep(aapbbp)1/(ap-bp),p=q≠0[ap-bpp(lna-lnb)]1/p,p≠0,q=0ab,p=q=0.The mean𝒜1=𝒜 is the arithmetic mean and the mean 𝒮1,0=ℒ is the logarithmic mean. As Carlson remarked in [5], the logarithmic mean can be represented also by
(1.9)L(a,b)=a-b2tanh-1⁡((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
(1.10)SB(a,b)={b2-a2cos⁡-1⁡(a/b),ifa<ba2-b2cosh-1(a/b),ifa>b
(see [6, 7]). It has been established in [3] that
(1.11)L=SB(A,G),P=SB(G,A),T=SB(A,A2),M=SB(A2,A).


## 2.Interlacing Property of Power Means

Given two meansM and N, we will write M<N if
(2.1)M(a,b)<N(a,b),fora≠b.It is known that the family of power means is an increasing family of means, thus(2.2)Ap<Aq,ifp<q.
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)k(x,y)={|x|-|y|x-y,x≠ysign(x),x=y,l(x,y)={ℒ(x,y),x>0,y>00,x≥0,y≥0,xy=0.Theorem 2.1.
Letp,q,r,s∈ℝ. Then the comparison inequality
(2.4)Sp,q≤Sr,s
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:
(2.5)ab23<SB(a,b)<a+2b3,a≠b.Being rather complicated, the Seiffert-type means were evaluated by simpler means, first of all by power means. Theevaluation of a given mean M by power means assumes the determination of some real indices p and q such that 𝒜p<M<𝒜q. 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 𝒜r if p<r<q.For the logarithmic mean in [10], it was determined the optimal evaluation
(2.6)A0<L<A1/3.
For the Seiffert means, there are known the evaluations
(2.7)A1/3<P<A2/3,
proved in [11] and
(2.8)A1<T<A2,
given in [2]. It is also known that
(2.9)A1<M<T,
as it was shown in [3]. Moreover in [12] it was determined the optimal evaluation
(2.10)Aln⁡⁡2/ln⁡⁡π<P<A2/3.
Using these results we deduce the following chain of inequalities:
(2.11)A0<L<A1/2<P<A1<M<T<A2.
To prove the full interlacing property of power means, our aim is to show that 𝒜3/2 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.


## 3.Main Results

We add to the inequalities (2.11) the next results.Theorem 3.1.
The following inequalities(3.1)M<A3/2<T
are satisfied.Proof.
First of all, let us remark that𝒜3/2=𝒮3,3/2. So, for the first inequality in (3.1), it is sufficient to prove that the following chain of inequalities
(3.2)M<A2+2A3<S3,1<S3,3/2
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
(3.3)1+t2+23<3+t23,
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
(3.4)S3,3/2<S4,1=AA223<T.
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).Corollary 3.2.
The following two-sided inequality(3.5)xsinh⁡-1⁡x<A3/2(1-x,1+x)<xtan-1⁡x,
is valid for all 0<x<1.



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*Source: 430692-2012-07-03.xml*