When is a family of generalized means a scale?

Mathematics – Functional Analysis

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

Scientific paper

For a family {k_t | t \in I} of real C^2 functions defined on U (I, U -- open intervals) and satisfying some mild regularity conditions, we prove that the mapping I \ni t --> k_t^{-1}(\sum_{i=1}^n w_i k_t(a_i)) is a continuous bijection between I and (min a, max a), for every fixed non-constant sequence a = (a_i)_{i=1}^n with values in U and every set, of the same cardinality, of positive weights w=(w_i)_{i=1}^n. In such a situation one says that the family of functions {k_t} generates a scale on U. The precise assumptions in our result read (all indicated derivatives are with respect to x \in U) (i) k'_t does not vanish anywhere in U for every t \in I, (ii) I \ni t \mapsto \frac{k"_t(x)}{k'_t(x)} is increasing, 1--1 on a dense subset of U and onto the image R for every x \in U. This result makes possible few new things as well as new proofs of classical results.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

When is a family of generalized means a scale? does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with When is a family of generalized means a scale?, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and When is a family of generalized means a scale? will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-346835

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.