Support-type properties of generalized convex functions

Wa̧sowicz, Szymon

doi:10.1016/j.jmaa.2009.10.070

Cited by 3 publications

(2 citation statements)

References 14 publications

(24 reference statements)

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“…Observe that Theorem 2 gives the well-known support property for the classical convex functions in the particular setting m = 1; moreover, together with Theorem 1, generalize a particular case of Wąsowicz's result [46][47][48]. In a recent paper, the same author has presented a complete solution of generalized support problems in Chebyshev's setting [50].…”

Section: Support Properties For Generalized Convex Functionsmentioning

confidence: 53%

The Hermite–Hadamard inequality in Beckenbach's setting

Bessenyei

2010

Journal of Mathematical Analysis and Applications

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The classical Hermite-Hadamard inequality gives a lower and an upper estimations for the integral average of convex functions defined on compact intervals, involving the midpoint and the endpoints of the domain. The aim of the present paper is to extend this inequality and to give analogous results when the convexity notion is induced by Beckenbach families.The key tool of the investigations is based on some general support theorems that are obtained via the pure geometric properties of Beckenbach families and can be considered as generalizations of classical support and chord properties of ordinary convex functions. The Markov-Krein-type representation of Beckenbach families is also investigated.

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Section: Support Properties For Generalized Convex Functionsmentioning

confidence: 53%

The Hermite–Hadamard inequality in Beckenbach's setting

Bessenyei

2010

Journal of Mathematical Analysis and Applications

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“…The author reproved these inequalities by a support technique (cf. [11,13]). If [−1, 1] ⊂ I and x = −1, y = 1, then inequalities (4), (5), (6) reduce to…”

Section: Introductionmentioning

confidence: 99%

Inequalities between remainders of quadratures

Komisarski

Wa̧sowicz

2017

Aequat. Math.

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It is well-known that in the class of convex functions the (nonnegative) remainder of the Midpoint Rule of the approximate integration is majorized by the remainder of the Trapezoid Rule. Hence the approximation of the integral of the convex function by the Midpoint Rule is better than the analogous approximation by the Trapezoid Rule. Following this fact we examine remainders of certain quadratures in the classes of convex functions of higher orders. Our main results state that for 3-convex (5-convex, respectively) functions the remainder of the 2-point (3-point, respectively) Gauss quadrature is non-negative and it is not greater than the remainder of the Simpson's Rule (4-point Lobatto quadrature, respectively). We also check the 2-point Radau quadratures for 2-convex functions to demonstrate that similar results fail to hold for convex functions of even orders. We apply Peano Kernel Theorem as a main tool of our considerations.

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New integral representations of nth order convex functions

Rajba

2011

Journal of Mathematical Analysis and Applications

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Support-type properties of generalized convex functions

Cited by 3 publications

References 14 publications

The Hermite–Hadamard inequality in Beckenbach's setting

The Hermite–Hadamard inequality in Beckenbach's setting

Inequalities between remainders of quadratures

New integral representations of nth order convex functions

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