Hermite?Hadamard inequalities for generalized convex functions

Bessenyei, Mihály

doi:10.1007/s00010-004-2730-1

Cited by 24 publications

(51 citation statements)

References 48 publications

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“…If x 1 is a boundary point of I it need not to be the case. Observe that a convex functionf (x) = − √ 1 − x 2 , x ∈ [−1,1], has no affine support both at x 1 = −1 and at x 1 = 1.…”

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confidence: 99%

Support-type properties of convex functions of higher order and Hadamard-type inequalities

Wa̧sowicz

2007

Journal of Mathematical Analysis and Applications

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It is well known that every convex function f : I → R (where I ⊂ R is an interval) admits an affine support at every interior point of I (i.e. for any x 0 ∈ Int I there exists an affine function a : I → R such that a(x 0 ) = f (x 0 ) and a f on I ). Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree no greater than the order of convexity. In this paper the attaching method is developed. It is applied to obtain the general result-Theorem 2, from which the mentioned above support theorem and some related properties of convex functions of higher (both odd and even) order are derived. They are applied to obtain some known and new Hadamard-type inequalities between the quadrature operators and the integral approximated by them. It is also shown that the error bounds of quadrature rules follow by inequalities of this kind.

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confidence: 99%

Support-type properties of convex functions of higher order and Hadamard-type inequalities

Wa̧sowicz

2007

Journal of Mathematical Analysis and Applications

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“…3,4]). Let (ω 0 , ω 1 ) be a positive Chebyshev system over I , J ⊂ I be a proper subinterval and f : J → R. Then the following statements are equivalent:…”

Section: Introductionmentioning

confidence: 96%

“…On the other hand, we can always assume that ω 0 is a positive function, because for every Chebyshev system (ω 0 , ω 1 ), there exists α, β ∈ R such that αω 0 + βω 1 > 0 (cf. [2][3][4]). In the sequel, for fixed x, y ∈ I , the partial functions u  → Ω (u, y) and u  → Ω (x, u) will be denoted by Ω (·, y) and Ω (x, ·), respectively.…”

Section: Introductionmentioning

confidence: 96%

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Korovkin type theorems and approximate Hermite–Hadamard inequalities

Makó

2012

Journal of Approximation Theory

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The main results of this paper offer sufficient conditions in order that an approximate lower Hermite-Hadamard type inequality implies an approximate convexity property. The failure of such an implication with constant error term shows that functional error terms should be considered for the inequalities and convexity properties in question. The key for the proof of the main result is a Korovkin type theorem which enables us to deduce the approximate convexity property from the approximate lower Hermite-Hadamard type inequality via an iteration process.

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“…We would like to refer the reader to [10,2,16,21,19,13,1,22,6,3] and references therein for more information. In this paper we refine one side of H-H inequality in NPC global spaces via sequences.…”

Section: Introductionmentioning

confidence: 99%