Interval-type theorems concerning means

Paweł Pasteczka

Abstract


Each family M of means has a natural, partial order (point-wise order), that is M ≤ N iff M(x) ≤ N(x) for all admissible x.  
In this setting we can introduce the notion of interval-type set (a subset I ⊂ M such that whenever M ≤  P ≤ N for some M,N ∈ I and P ∈ M then P ∈ I). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered. 
In the present paper we consider this property for Gini means and Hardy means. Moreover some results concerning L∞ metric among (abstract) means will be obtained.

Keywords


means; squeeze theorem; sandwich theorems; distance between means; Hardy means

Mathematics Subject Classification (2010)


26E60; 26D15

References


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