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Revista Colombiana de Matemáticas

versão impressa ISSN 0034-7426

Rev.colomb.mat. vol.52 no.2 Bogotá jul./dez. 2018

http://dx.doi.org/10.15446/recolma.v52n2.77157 

Original articles

Sandwich theorem for reciprocally strongly convex functions

Teorema del Sandwich para funciones fuerte-recíprocamente convexas

Mireya Bracamonte1  *  , José Giménez2  , Jesús Medina3 

1 Escuela Superior Politécnica del Litoral (ESPOL) Departamento de Matemáticas, Facultad de Ciencias Naturales y Matemática, Km 30.5 Vía Perimetral, Campus Gustavo Galindo, Guayaquil, Ecuador. e-mail: mirebrac@gmail.com

2 Universidad de los Andes, Departamento de Matemáticas, Facultad de Ingeniería. Mérida, Venezuela. e-mail: jgimenez@ula.ve

3 Universidad Centroccidental Lisandro Alvarado, Departamento de Matemáticas, Decanato de Ciencias y Tecnología. Barquisimeto, Venezuela. e-mail: jesus.medina@ucla.edu.ve

Abstract

We introduce the notion of reciprocally strongly convex functions and we present some examples and properties of them. We also prove that two real functions f and g, defined on a real interval [a, b], satisfy

for all x, y ∈ [a, b] and t ∈ [0, 1] iff there exists a reciprocally strongly convex function h: [a, b] → R such that f (x) ≤ h(x) ≤ g(x) for all x ∈ [a, b].

Finally, we obtain an approximate convexity result for reciprocally strongly convex functions; namely we prove a stability result of Hyers-Ulam type for this class of functions.

Key words: Convex functions; Sandwich theorem; Hyers-Ulam

Resumen

En este artículo introducimos la noción de funciones recíproca-fuertemente convexas y presentamos algunos ejemplos y propiedades. Además se demuestran que dos funciones f y g, definidas en el intervalo real [a, b] satisfacen la desigualdad

para todo x, y ∈ [a, b] y t ∈ [0, 1] si, y sólo si, existe una función recíproca-fuertemente convexa h : [a, b] → R tal que f (x) ≤ h(x) ≤ g (x) para todo x ∈ [a, b].

Finalmente, se obtiene un resultado de aproximación convexa para esta clase de funciones.

Palabras-clave: Funciones convexas; Teorema del Sandwich; Hyers-Ulam

Text complete end PDF

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Received: January 31, 2018; Accepted: May 08, 2018

* Correspondencia: Mireya Bracamonte, Departamento de Matemáticas, Facultad de Ciencias Naturales y Matemática, Escuela Superior Politécnica del Litoral (ESPOL), Km 30.5 Vía Perimetral, Campus Gustavo Galindo Guayaquil, Ecuador. Correo electrónico: mirebrac@gmail.com.

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