The notion of strongly n-convexfunctions with modulus c > 0 is introduced and investigated. Relationships between such functions and n-convexfunctions in the sense of Popoviciu as well as generalized convex funct...
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The notion of strongly n-convexfunctions with modulus c > 0 is introduced and investigated. Relationships between such functions and n-convexfunctions in the sense of Popoviciu as well as generalized convexfunctions in the sense of Beckenbach are given. Characterizations by derivatives are presented. Some results on strongly Jensen n-convexfunctions are also given. (C) 2010 Elsevier Ltd. All rights reserved.
The classes of n-Wright-convexfunctions and n-Jensen-convexfunctions are compared with each other. It is shown that for any odd natural number n the first one is the proper subclass of the second one. To reach this ...
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The classes of n-Wright-convexfunctions and n-Jensen-convexfunctions are compared with each other. It is shown that for any odd natural number n the first one is the proper subclass of the second one. To reach this aim new tools connected with measure theory are developed. (C) 2012 Elsevier Inc. All rights reserved.
We present a computer program solving inequalities of the form a1f(alpha 1x+(1-alpha 1)y)+& ctdot;+anf(alpha nx+(1-alpha n)y)yf(t)\, dt, $$\end{document}where the unknown function f:R -> R\documentclass[12pt]{m...
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We present a computer program solving inequalities of the form a1f(alpha 1x+(1-alpha 1)y)+& ctdot;+anf(alpha nx+(1-alpha n)y)<= 1y-x integral xyf(t)dt,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ a_1f(\alpha _1x+(1-\alpha _1)y)+\cdots +a_nf(\alpha _nx+(1-\alpha _n)y)\leqslant \fracconvex{y-x}\int _x<^>yf(t)\, dt, $$\end{document}where the unknown function f:R -> R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\mathbb R\rightarrow \mathbb R$$\end{document} is assumed to be continuous. This inequality includes, in particular cases, many well-known inequalities such as the classical Hermite-Hadamard inequality, the Hermite-Hadamard inequalities of higherorders, the Bullen inequality, and others. In the construction of our program, we use three theoretical results. The first one is used to show that every continuous solution of the inequality is a convex function of some order. The second one is a simple sufficient condition under which every convex function of this order satisfies the inequality we are considering. If this condition fails, the program checks a more complicated condition which is necessary and sufficient for the inequality to be satisfied by every convex function of that order. If this third condition is satisfied, then our inequality is solved completely;in the opposite case, the exact form of the solutions of the inequality in question remains unknown. However, if this is the case, we know that the quadrature in question is not definite. In the paper, we provide many examples that would be hard to calculate by hand. In the list of references, we listed many papers where computer programs w
In the approximate integration some inequalities between the quadratures and the integrals approximated by them are called extrernalities. On the other hand, the set of all quadratures is convex. We are trying to find...
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In the approximate integration some inequalities between the quadratures and the integrals approximated by them are called extrernalities. On the other hand, the set of all quadratures is convex. We are trying to find possible connections between extrernalities and extremal quadratures (in the sense of extreme points of a convex set). Of course, the quadratures are the integrals with respect to discrete measures and, moreover, a quadrature is extremal if and only if the associated measure is extremal. Hence the natural problem arises to give some description of extremal measures with prescribed moments in the general (not only discrete) case. In this paper we deal with symmetric measures with prescribed first four moments. The full description (with no symmetry assumptions, and/or not only four moments are prescribed and so on) is far to be done. (C) 2014 Elsevier Inc. All rights reserved.
We provide a generalization of a recent result of Anastassiou related to the well-known Ostrowski inequality, as well as some related results. Our results subsume, extend, and consolidate a number of known results.
We provide a generalization of a recent result of Anastassiou related to the well-known Ostrowski inequality, as well as some related results. Our results subsume, extend, and consolidate a number of known results.
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