On approximately t-convex functions

Attila Házy, Zsolt Páles

Research output: Contribution to journalArticle

29 Citations (Scopus)

Abstract

A real valued function / defined on an open convex set D is called (ε, p, t)-convex if it satisfies f (tx + (1 - t)y) ≤ t f (x) + (1 - t)f(y) + ∑ i=0 kε i|x-y| pi for x, y ∈ D, where epsi; = (epsi; 0,...,epsi; k) ∈ [0,∞[ k+1, p = (P 0, ...,p k) ∈[0, 1[ k+1 and t ∈]0, 1[are fixed parameters. The main result of the paper states that if / is locally bounded from above at a point of D and (epsi;, p, t)-convex then it satisfies the convexity-type inequality f(sx + (1 - s)y) ≤ s f(x) + (1 - s)f(y) + ∑ i=0 kepsi; iφ pi,t(s)|x - y| pi for x, y epsi; D, s epsi; [0, 1], where φ pi,t : [0, 1] → ℝ is defined by φ ph,t(s) = max {1/(1 - t) pi - (1 - t);1/ t pi - t} (s(1 - s)) pi. The particular case k = 0, p = 0 of this result is due to PÁLES [Pál00], the case k = 0, p = 0 and t = 1/2 was investigated by NG and NIKODEM [NN93]. The specialization k - 0, epsi; 0 = 0 yields the celebrated theorem of BERNSTEIN and DOETSCH [BD15]. The case k = 1, epsi; = (epsi; 0,epsi; 1), P = (1,0) and t = 1/2 was investigated in HÁZY and PÁLES [HP04].

Original languageEnglish
Pages (from-to)489-501
Number of pages13
JournalPublicationes Mathematicae
Volume66
Issue number3-4
Publication statusPublished - Jun 16 2005

    Fingerprint

Keywords

  • (epsi;, p)-midconvexity
  • (epsi;, p, t)-convexity
  • Bernstein-Doetsch theorem

ASJC Scopus subject areas

  • Mathematics(all)

Cite this