### Abstract

A family F of continuous real functions defined on an interval I is a twoparameter family if for any two points (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ I × ℝ with x_{1} ≠ x_{2} there exists exactly one ϕ = ϕ_{(x1,y1)(x2,y2)} ∈ F such that ϕ(x_{i}) = y_{i} for i = 1, 2. Following Beckenbach [1] a function f: I → ℝ is said to be F-convex if for any x_{1}, x_{2} ∈ I, x_{1} < x_{2} f(x) ≤ ϕ_{(x1,f (x1))(x2,f (x2))}(x), x_{1} ≤ x ≤ x_{2}. In the talk some characterization of pairs of functions that can be separated by a function belonging to F are given. As a consequence, the following sandwich theorem and a Hyers-Ulam-type stability result are obtained:.

Original language | English |
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Pages (from-to) | 39-40 |

Number of pages | 2 |

Journal | Real Analysis Exchange |

Volume | 32 |

Issue number | 1 |

Publication status | Published - Jan 1 2007 |

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### ASJC Scopus subject areas

- Analysis
- Geometry and Topology

### Cite this

*Real Analysis Exchange*,

*32*(1), 39-40.

**Generalized covex functions and separation theorems.** / Nikodem, Kazimierz; Páles, Z.

Research output: Contribution to journal › Article

*Real Analysis Exchange*, vol. 32, no. 1, pp. 39-40.

}

TY - JOUR

T1 - Generalized covex functions and separation theorems

AU - Nikodem, Kazimierz

AU - Páles, Z.

PY - 2007/1/1

Y1 - 2007/1/1

N2 - A family F of continuous real functions defined on an interval I is a twoparameter family if for any two points (x1, y1), (x2, y2) ∈ I × ℝ with x1 ≠ x2 there exists exactly one ϕ = ϕ(x1,y1)(x2,y2) ∈ F such that ϕ(xi) = yi for i = 1, 2. Following Beckenbach [1] a function f: I → ℝ is said to be F-convex if for any x1, x2 ∈ I, x1 < x2 f(x) ≤ ϕ(x1,f (x1))(x2,f (x2))(x), x1 ≤ x ≤ x2. In the talk some characterization of pairs of functions that can be separated by a function belonging to F are given. As a consequence, the following sandwich theorem and a Hyers-Ulam-type stability result are obtained:.

AB - A family F of continuous real functions defined on an interval I is a twoparameter family if for any two points (x1, y1), (x2, y2) ∈ I × ℝ with x1 ≠ x2 there exists exactly one ϕ = ϕ(x1,y1)(x2,y2) ∈ F such that ϕ(xi) = yi for i = 1, 2. Following Beckenbach [1] a function f: I → ℝ is said to be F-convex if for any x1, x2 ∈ I, x1 < x2 f(x) ≤ ϕ(x1,f (x1))(x2,f (x2))(x), x1 ≤ x ≤ x2. In the talk some characterization of pairs of functions that can be separated by a function belonging to F are given. As a consequence, the following sandwich theorem and a Hyers-Ulam-type stability result are obtained:.

UR - http://www.scopus.com/inward/record.url?scp=84865309326&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865309326&partnerID=8YFLogxK

M3 - Article

VL - 32

SP - 39

EP - 40

JO - Real Analysis Exchange

JF - Real Analysis Exchange

SN - 0147-1937

IS - 1

ER -