### 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 |
---|---|

Pages (from-to) | 78-91 |

Number of pages | 14 |

Journal | Acta Mathematica Hungarica |

Volume | 132 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jul 2011 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Acta Mathematica Hungarica*,

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

**Strengthening of strong and approximate convexity.** / Makó, J.; Páles, Z.

Research output: Contribution to journal › Article

*Acta Mathematica Hungarica*, vol. 132, no. 1-2, pp. 78-91. https://doi.org/10.1007/s10474-010-0056-0

}

TY - JOUR

T1 - Strengthening of strong and approximate convexity

AU - Makó, J.

AU - Páles, Z.

PY - 2011/7

Y1 - 2011/7

N2 - 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.

AB - 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.

KW - (E,t)-convexity

KW - 39B12

KW - 39B22

KW - approximate and strong convexity

UR - http://www.scopus.com/inward/record.url?scp=79957909374&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79957909374&partnerID=8YFLogxK

U2 - 10.1007/s10474-010-0056-0

DO - 10.1007/s10474-010-0056-0

M3 - Article

VL - 132

SP - 78

EP - 91

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 1-2

ER -