Generalized covex functions and separation theorems

Kazimierz Nikodem, Z. Páles

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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:.

Original languageEnglish
Pages (from-to)39-40
Number of pages2
JournalReal Analysis Exchange
Volume32
Issue number1
Publication statusPublished - Jan 1 2007

Fingerprint

Separation Theorem
Generalized Functions
Sandwich
Interval
Theorem
Family

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

Cite this

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

In: Real Analysis Exchange, Vol. 32, No. 1, 01.01.2007, p. 39-40.

Research output: Contribution to journalArticle

Nikodem, Kazimierz ; Páles, Z. / Generalized covex functions and separation theorems. In: Real Analysis Exchange. 2007 ; Vol. 32, No. 1. pp. 39-40.
@article{607dcbf5ced14e929b32d3b7357cca55,
title = "Generalized covex functions and separation theorems",
abstract = "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:.",
author = "Kazimierz Nikodem and Z. P{\'a}les",
year = "2007",
month = "1",
day = "1",
language = "English",
volume = "32",
pages = "39--40",
journal = "Real Analysis Exchange",
issn = "0147-1937",
publisher = "Michigan State University Press",
number = "1",

}

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 -