### 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} ^{k}ε _{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} ^{k}epsi; _{i}φ _{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 language | English |
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Pages (from-to) | 489-501 |

Number of pages | 13 |

Journal | Publicationes Mathematicae |

Volume | 66 |

Issue number | 3-4 |

Publication status | Published - Jun 16 2005 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*66*(3-4), 489-501.