A characterization of Beckenbach families admitting discontinuous Jensen affine functions

• Autores: Michal Lewicki
• Localización: Publicationes Mathematicae Debrecen, ISSN 0033-3883, Tomus 81, Fasc. 1-2, 2012, págs. 65-80
• Idioma: inglés
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• Resumen

  • Let F : R3 ¡æ R be a continuous function such that F := {R ¡õ x 7¡æ F(x; a; b) ¡ô R : a; b ¡ô R} is a Beckenbach family. Additionally, we assume that for each a; b ¡ô R the functions R ¡õ x 7¡æ F(x; a; b) ¡ô R are monotonic. We show that if there exists a function which is discontinuous at some point and Jensen affine with respect to the family F, then there exists a strictly increasing and continuous function h : R ¡æ R and continuous G;H : R2 ¡æ R such that F(u; a; b) = h(G(a; b)u + H(a; b)); (.) for all u; a; b ¡ô R. As a consequence we get an independent proof of theorem of J. Matkowski. Finally, we characterize Beckenbach families of the form (.).