Description

• Title:
  Generalization of the harmonic weighted mean via Pythagorean invariance identity and application
• Creator:
  Kahlig, Peter;
  Matkowski, Janusz
• Description:
  Under some simple conditions on the real functions f and g defined on an interval I⊂(0,∞), the two-place functions Af(x; y) = f (x)+y-f (y) and Gg(x; y) = \frac{g(x)}{g(y)}y generalize, respectively, A and G, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity G ◦ (H,A) = G (equivalent to the Pythagorean harmony proportion), a suitableweighted extension Hf,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of Hf,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.
• Object availability:
  free access
• Date:
  2020.07.09
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Annales Mathematicae Silesianae; Vol. 34 No. 1 (2020); 104-122;
  Annales Mathematicae Silesianae; Tom 34 Nr 1 (2020); 104-122;
  2391-4238;
  0860-2107
• Format:
  application/pdf
• Publisher:
  University of Silesia Press
• Relation: https://journals.us.edu.pl/index.php/AMSIL/article/view/13636/10442
• Subject:
  generalized arithmetic and geometric means;
  invariance identity;
  generalized harmonic mean;
  functional equations;
  mean-type mappings;
  iteration;
  convergence of iterates;
  invariant functions
• Identifier: https://journals.us.edu.pl/index.php/AMSIL/article/view/13636
• Data provider:
  Czasopisma Wydawnictwa Uniwersytetu Śląskiego
• Can I use it?:
  seek permission
• Type:
  other