### Abstract

Let X be a real linear space and D {subset double equals} X be a nonempty convex subset. Given an error function E:[0,1]×(D-D)→ℝ∪{+∞} and an element t ∈]0, 1[, a function f:D→ℝ is called (E,t)-convex if f(tx+(1-t)y)≦ tf(x)+(1-t)f(y)+E(t,x-y) for all x,y∈D. The main result of this paper states that, for all a,b∈(ℕ∪{0})+{0,t,1-t} such that {a,b,a+b}∩ℕ≠, every (E,t)-convex function is also, convex, where, As a consequence, under further assumptions on E, the strong and approximate convexity properties of (E,t)-convex functions can be strengthened.

Original language | English |
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Pages (from-to) | 78-91 |

Number of pages | 14 |

Journal | Acta Mathematica Hungarica |

Volume | 132 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jul 1 2011 |

### Keywords

- (E,t)-convexity
- 39B12
- 39B22
- approximate and strong convexity

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Makó, J., & Páles, Z. S. (2011). Strengthening of strong and approximate convexity.

*Acta Mathematica Hungarica*,*132*(1-2), 78-91. https://doi.org/10.1007/s10474-010-0056-0