On $\varphi$-convexity

Makó, Judit

doi:10.5486/pmd.2012.4930

Cited by 10 publications

(3 citation statements)

References 24 publications

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“…Many authors examined this inequality from many contexts. For example (Házy, 2005 and, (Házy et al, 2004(Házy et al, , 2005(Házy et al, and 2009, (Makó et al, 2010(Makó et al, , 2011(Makó et al, , 2012(Makó et al, and 2013, (Ng et al, 1993), (Páles, 2003), (Tabor et al, 2009a(Tabor et al, , 2009b(Tabor et al, , 2010a(Tabor et al, , 2010b). If 𝛼 is constant zero, we have the notion of classical Jensen-convexity.…”

Section: Introductionmentioning

confidence: 99%

Counter-examples to Breckner-convexity

Házy,

Makó

2023

MDT

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In this paper, we examine convexity type inequalities. Let D be a nonempty convex subset of a linear space, c>0 and α:D-D→R be a given even function. The inequality f((x+y)/2) ≤ c f(x) + c f(y) + α(x-y) (x,y € D) is the focus of our examinations. We will construct an example to show that for c=1, this Jensen type inequality does not imply the convexity of the function. Then, we compare this inequality with Hermite–Hadamard type inequalities.

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Section: Introductionmentioning

confidence: 99%

Counter-examples to Breckner-convexity

Házy,

Makó

2023

MDT

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“…With the help of error function Φ, we can introduce terminologies such as approximately monotone, Hölder, convex and affine functions. These function classes are studied in depth in the papers [2]- [4], [6], where along with structural properties, some characterizations, decompositions, several sandwich type theorems and applications are discussed. For readability purposes, we recall these notions.…”

Section: Introductionmentioning

confidence: 99%

“…The following definition was introduced in [6]. A function f : I → R is said to be Φ-convex if the functional inequality…”

Section: Introductionmentioning

confidence: 99%

Hermite–Hadamard-Type Inequalities and Two-Point Quadrature Formula

Barić¹

2022

Mathematics

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As convexity plays an important role in many aspects of mathematical programming, e.g., for obtaining sufficient optimality conditions and in duality theorems, and one of the most important inequalities for convex functions is the Hermite–Hadamard inequality, the importance of this paper lies in providing some new improvements for convex functions and new directions in studying new variants of the Hermite–Hadamard inequality. The first part of the article includes some known concepts regarding convex functions and related inequalities. In the second part of the study, a derivation of the Hermite–Hadamard inequality for convex functions of higher order is given, emphasizing the purpose and importance of some quadrature formulas. In the third section, the applications of the main results are presented by obtaining Hermite–Hadamard-type estimates for various classical quadrature formulas such as the Gauss–Legendre two-point quadrature formula and the Gauss–Chebyshev two-point quadrature formulas of the first and second kind.

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Approximate convexity with the standard deviation’s error

Żołdak

2015

Aequat. Math.

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On $\varphi$-convexity

Cited by 10 publications

References 24 publications

Counter-examples to Breckner-convexity

Counter-examples to Breckner-convexity

Hermite–Hadamard-Type Inequalities and Two-Point Quadrature Formula

Approximate convexity with the standard deviation’s error

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