On a Hierarchy of Means

For a class of partially ordered means we introduce a notion of the (nontrivial) cancelling mean. A simple method is given which helps to determine cancelling means for well known classes of Holder and Stolarsky means.


Introduction
A mean is a map M : R + × R + → R + , with a property min(a, b) ≤ M (a, b) ≤ max(a, b), for each a, b ∈ R + .
Denote by Ω the class of means which are symmetric (in variables a, b), reflexive and homogeneous (necessarily of order one). We shall consider in the sequel only means from this class.
The set of means can be equipped with a partial ordering defined by M ≤ N if and only if M (a, b) ≤ N (a, b) for all a, b ∈ R + . Thus, ∆ is an ordered family of means if for any M, N ∈ ∆ we have M ≤ N or N ≤ M .
Most known ordered family of means is the following family ∆ 0 of elementary means, It is well known that the inequality A s (a, b) < A t (a, b) holds for all a, b ∈ R + , a = b if and only if s < t. This property is used in a number of papers for approximation of a particular mean by means from the class {A s }.
Hence (cf [3], [4], [10]), where all bounds are best possible and Seiffert means P and T are defined by In the recent paper [8] we introduce a more complex structured class of means {λ s }, given by Those means are obviously symmetric and homogeneous of order one. We also proved that λ s is monotone increasing in s ∈ R; therefore {λ s } represents an ordered family of means.
Among others, the following approximations are obtained for a = b: λ −4 < H < λ −3 ; λ −1 < G < λ −1/2 ; λ 0 < L < λ 1 < I < λ 2 = A; λ 5 < S, and there is no finite s > 5 such that the inequality S(a, b) ≤ λ s (a, b) holds for each a, b ∈ R + . This last result shows that, in a sense, the mean S is "greater" than any other mean from the class {λ s }. We shall say that S is the cancelling mean for the class {λ s }.

Definition 1
The mean S * (∆) is right cancelling mean for an ordered class of means ∆ ⊂ Ω if there exists M ∈ ∆ such that S * (a, b) ≥ M (a, b) but there is no mean N ∈ ∆ such that the inequality N (a, b) ≥ S * (a, b) holds for each a, b ∈ R + .
Definition of the left cancelling mean S * is analogues. Of course that the left and right cancelling means exist for arbitrary ordered family of means as S * (a, b) = max(a, b), S * (a, b) = min(a, b). We call them trivial.
The aim of this article is to determine non-trivial cancelling means for some well known classes of ordered means. We shall also give a simple criteria for the right cancelling mean with further discussion in the sequel.
As an illustration of problems and methods which shall be treated in this paper, we prove firstly the following,

Cancellation theorem for the Generalized Logarithmic Means
The family of Generalized Logarithmic Means {L p } is given by It is a subclass of well-known Stolarsky means (cf [2], [5], [7]) hence symmetric, homogeneous and monotone increasing in p. Therefore it represents an ordered family of means.
with those bounds as best possible.
Proof We prove firstly that the inequality L 3 (a, b) < A(a, b) holds for all a, b ∈ R + , a = b.

Indeed,
, Thus p = 3 is the largest p such that the inequality L p (a, b) ≤ A(a, b) holds for each a, b ∈ R + , since for p > 3 and t sufficiently small (i.e., a is sufficiently close to b) we have that L p (a, b) > A(a, b).
We shall show now that A is the right cancelling mean for the class {L p }.
Indeed, since lim t→1 − L p p A p = 0 for fixed p, p > 3, we conclude that the inequality L p ≥ A cannot hold. Hence by Definition 1., A is the right cancelling mean for the class {L p }.
Noting that H(a, b) = ab A(a,b) and L −p (a, b) = ab Lp(a,b) , we readily get 2. Characteristic number and characteristic function Let M = M (a, b) be an arbitrary homogeneous and symmetric mean. In order to facilitate determination of a non-trivial right cancelling mean, we introduce here a notion of characteristic number σ(M ) as Because of homogeneousness, we have and the result follows. Therefore, Some simple reasoning gives the next, This assertion is especially important in applications.
In this way comparison between means reduces to comparison between their characteristic functions ( [8], [10], [11]). Obviously, ; φ A (t) = 1; We shall give now some applications of the above.  Another consequence is the cancellation assertion for the family of Hölder means A r = A r (a, b) := (A(a r , b r )) 1/r = ( a r +b r 2 ) 1/r , A 0 = G. Since [log A(x, y)] xy = − 1 (x+y) 2 < 0, we obtain (as is already stated) that A r are monotone increasing with r.
Therefore, the inequality A 2 (a, b) ≤ S(a, b) holds for all a, b ∈ R + .
Also, since we conclude that r = 2 is best possible upper bound for A r ≤ S to hold.
Values for S * (A r ) and S * (A r ) follow from Theorem 2.1.

Cancellation theorem for the class of Stolarsky means
There is a plenty of papers (cf [2], [5], [7]) studying different properties of the so-called Stolarsky (or extended) two-parametric mean value, defined for positive values of x, y, x = y by the following x, y = x > 0.
In this form it was introduced by K. Stolarsky in [5].
Main properties of Stolarsky means are given in the following assertion. c. homogeneous of order one, that is I r,s (tx, ty) = tI r,s (x, y), t > 0; d. monotone increasing in either r or s; e. monotone increasing in either x or y; f. logarithmically convex for r, s ∈ R − and logarithmically concave for r, s ∈ R + .
where given constants are best possible.
Proof We prove firstly that I 3,3 (a, b) ≤ S(a, b) and that s = 3 is the largest constant such that the inequality I s,s (a, b) ≤ S(a, b) holds for all a, b ∈ R + . For this aim we need a notion of Lehmer means l r defined by They are continuous and strictly increasing in r ∈ R (cf [11]). We also need the following interesting identity which is new to our modest knowledge.
Proof Indeed, by the definition of I s,s , we get Now, putting s = 3 in the above identity and applying Lemma 3.3, the proof follows immediately.
Therefore, the mean S is not an exclusive right cancelling mean in the above assertions. Moreover, we can construct a whole class of means which may replace the mean S as the right cancelling mean. Proof We shall prove first that K r is monotone increasing in r ∈ R. For this aim, consider the weighted arithmetic mean A p,q (x, y) := px + qy, where p, q are arbitrary positive numbers such that p + q = 1. Since [log A p,q (x, y)] xy = − pq (px + qy) 2 , we conclude thatÃ r (p, q; a, b) := (pa r + qb r ) 1/r , is monotone increasing in r ∈ R.
Hence, the relationÃ r ( ; a, b) = K r (a, b), yields the proof. Now, since for fixed r > −1, and σ(K r ) = 2, it follows that K r is the right cancelling mean for the class {M s } analogously to the proof of Theorem 2.1.
Finally, we propose two open questions concerning the above matter.
Q2 Does there exists a non-trivial right cancelling mean for the class {K ′ r } ?