Optimality estimations for approximately midconvex functions

Tabor, Jacek; Żołdak, Marek

doi:10.1007/s00010-010-0033-2

Cited by 10 publications

(7 citation statements)

References 10 publications

(15 reference statements)

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“…Tabor, Jó. Tabor, and Żoldak [19,29,30,31,32] are generalized by Corollaries 4.4-4.7 to the vector-valued and set-valued setting. Similarly, using the explicit form of the function T 2 described in Remark 3.8, one can easily derive the results of Azócar, Gimenez, Nikodem and Sanchez [2] and Leiva, Merentes, Nikodem, and Sanchez [14] that are related to strongly K-Jensen convex real valued and set-valued functions from Corollary 4.5.…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

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: Resultsmentioning

confidence: 99%

“…Tabor, Jó. Tabor, and Żoldak [19,29,30,31,32]. For the set-valued and K-Jensen convex/concave setting more general statements will be formulated as direct consequences of our two main results below.…”

Section: Introductionmentioning

confidence: 94%

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, for p > 1 a function f : I → R satisfies (1) for some nonnegative ε if and only if f is nondecreasing. Another motivation for our paper comes from the theory of approximate convexity which has a rich literature, see for instance [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [36], [37], [38], [39]. In these papers several aspects of approximate convexity were investigated: stability problems, Bernstein-Doetsch-type theorems, Hermite-Hadamard type inequalities, etc.…”

Section: Introductionmentioning

confidence: 99%

On approximately monotone and approximately Hölder functions

Goswami,

Páles

2019

Preprint

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A real valued function f defined on a real open interval I is called Φ-monotone if, for all x, y ∈ I with x ≤ y it satisfies f (x) ≤ f (y) + Φ(y − x), where Φ : [0, ℓ(I)[ → R+ is a given nonnegative error function, where ℓ(I) denotes the length of the interval I. If f and −f are simultaneously Φ-monotone, then f is said to be a Φ-Hölder function.In the main results of the paper, we describe structural properties of these function classes, determine the error function which is the most optimal one. We show that optimal error functions for Φ-monotonicity and Φ-Hölder property must be subadditive and absolutely subadditive, respectively. Then we offer a precise formula for the lower and upper Φ-monotone and Φ-Hölder envelopes. We also introduce a generalization of the classical notion of total variation and we prove an extension of the Jordan Decomposition Theorem known for functions of bounded total variations.

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“…To present the following results we need the generalization of the notion of (ε, p)-midconvexity: In [13] the authors showed that to check the optimality of estimation (1.1) in the class of α-midconvex functions it suffices to check two inequalities. To quote this result it is convenient to formulate Condition T. Definition 1.3 (Condition T ).…”

Section: Introductionmentioning

confidence: 99%

“…In order to do that, a theorem from [13] and an algorithm based on interval arithmetic will be used.…”

Section: Introductionmentioning

confidence: 99%

Numerical verification of condition for approximately midconvex functions

Spurek

Tabor

2012

Aequat. Math.

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Let X be a normed space and V be a convex subset of X. Let α :It can be shown that every continuous α-midconvex function satisfies the following estimation:It is an important problem to verify for which functions α the above estimation is optimal. The conjecture of Páles that this is the case for functions of type α(r) = r p for p ∈ (0, 1), was proved by Makó and Páles (J Math Anal Appl 369:545-554, 2010). In this paper we present a computer assisted method to verify the optimality of this estimation in the class of piecewise linear functions α.Mathematics Subject Classification (2010). Primary 26A51; Secondary 39B62, 65G40.

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Optimality estimations for approximately midconvex functions

Cited by 10 publications

References 10 publications

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

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

On approximately monotone and approximately Hölder functions

Numerical verification of condition for approximately midconvex functions

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