Strengthening of strong and approximate convexity∗

Makó, Judit

doi:10.1007/s10474-010-0056-0

Cited by 9 publications

(8 citation statements)

References 15 publications

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“…In the next theorem and corollary, which are particular cases of the theorem in [6], the strong α-Jensen convexity property will be strengthened. We provide their proof here because it is much simpler and more transparent than in the general case.…”

Section: Strengthening the Strong Jensen Convexitymentioning

confidence: 99%

On strong (α,?d)-convexity

Makó¹,

Nikodem²

2012

Mathematical Inequalities &Amp; Applications

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Abstract. In this paper, strongly (α,T ) -convex functions, i.e., functions f : D → R satisfying the functional inequalityHere D is a convex set in a linear space, α is a nonnegative function on D − D , and T ⊆ R is a nonempty set. The main results provide various characterizations of strong (α,T ) -convexity in the case when T is a subfield of R .Mathematics subject classification (2010): Primary 39B62, 26B25.

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Section: Strengthening the Strong Jensen Convexitymentioning

confidence: 99%

On strong (α,?d)-convexity

Makó¹,

Nikodem²

2012

Mathematical Inequalities &Amp; Applications

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“…which proves that f is Φ γ -convex. Now define the sequence of error function Φ n by the iteration (25). By the assumption, f is Φ 1 = Φ-convex.…”

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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“…Observe that x = u 1 + • • • + u n−1 + (u n + (x − y)). Thus, using the inequality in (23), we arrive at…”

Section: φ-Monotone and φ-Hölder Envelopesmentioning

confidence: 99%

“…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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Strengthening of strong and approximate convexity∗

Cited by 9 publications

References 15 publications

On strong (α,?d)-convexity

On strong (α,?d)-convexity

On approximately convex and affine functions

On approximately monotone and approximately Hölder functions

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