Korovkin type theorems and approximate Hermite–Hadamard inequalities

Makó, Judit

doi:10.1016/j.jat.2012.05.010

Cited by 8 publications

(5 citation statements)

References 21 publications

(25 reference statements)

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“…Results, extending this approach to more general error terms and also to convexity concepts related to Chebyshev systems, have recently been obtained by Házy, Makó and Páles [7,8,10,11,15,16,17,18] and by Mureńko, Ja. Tabor, Jó.…”

Section: Introductionmentioning

confidence: 92%

Bernstein--Doetsch type theorems for set-valued maps of strongly and approximately convex and concave type

González¹,

Nikodem²,

Roa³

2014

Publ. Math. Debrecen

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In this paper, we investigate properties of set-valued mappings that establish connection between the values of this map at two arbitrary points of the domain and the value at their midpoint. Such properties are, for instance, Jensen convexity/concavity, K-Jensen convexity/concavity (where K is the set of nonnegative elements of an ordered vector space), and approximate/strong K-Jensen convexity/concavity. Assuming weak but natural regularity assumptions on the set-valued map, our main purpose is to deduce the convexity/concavity consequences of these properties in the appropriate sense. Our two main theorems will generalize most of the known results in this field, in particular the celebrated Bernstein-Doetsch Theorem from 1915, and thus they offer a unified view of these theories.

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Section: Introductionmentioning

confidence: 92%

Bernstein--Doetsch type theorems for set-valued maps of strongly and approximately convex and concave type

González¹,

Nikodem²,

Roa³

2014

Publ. Math. Debrecen

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“…Therefore, upon taking the infimum with respect to v ∈ [x, u[ and w ∈ ]u, y] in (27), it follows that…”

Section: Optimal Error Functionsmentioning

confidence: 99%

On approximately convex and affine functions

Goswami¹

2020

Preprint

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A real valued function f defined on a real open interval I is called Φ-convex if, for all x, y ∈ I, t ∈ [0, 1] it satisfieswhere Φ : R + → R + is a nonnegative error function. If f and −f are simultaneously Φ-convex, then f is said to be a Φ-affine function. In the main results of the paper, we describe the structural and inclusion properties of these two classes. We characterize these two classes of functions and investigate their relationship with approximately monotone and approximately-Hölder functions. We also introduce a subclass of error functions which enjoy the so-called Γ property and we show that the error function which is the most optimal for a Φ-convex function has to belong to this subclass. The properties of this subclass of error function are investigated as well. Then we offer two formulas for the lower Φ-convex envelop. Besides, a sandwich type theorem is also added.

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“…First, we recall a Korovkin type theorem, which will play an important role in the proof of the main result Theorem 11. For the historical background of these theorems, see the classical Korovkin theorem ( [Kor53], [AC94], [MP12]), which has a great importance in functional analysis.…”

Section: Lower Hermite-hadamard Type Inequalities For Schur-convex Fumentioning

confidence: 99%