Perturbed weighted trapezoid inequalities for convex functions with applications

Sever Silvestru Dragomir; Eder Kikianty

doi:10.56754/0719-0646.2603.507

Authors

Sever Silvestru Dragomir Applied Mathematics Research Group, ISILC, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia – School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa. https://orcid.org/0000-0003-2902-6805
Eder Kikianty Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield 0028, South Africa. https://orcid.org/0000-0003-4355-0339

DOI:

https://doi.org/10.56754/0719-0646.2603.507

Keywords:

Trapezoid inequality, midpoint inequaliy, Ostrowski’s inequality, Čebyšev’s inequality, norm inequality, semi-inner product

Abstract

We consider trapezoid type inequalities for twice differentiable convex functions, perturbed by a non-negative weight. Applications on a normed space \( (X, \lVert \,\cdot\, \rVert) \) are considered, by establishing bounds for the term
\[ \begin{multline*} \frac{1}{2} \left[\lVert \frac{x+y}{2} \rVert^p + \frac{\lVert x \rVert^p + \lVert y \rVert^p}{2} \right] - \int_{0}^{1} \lVert (1-t)x + ty \rVert^p \, dt, \\ x, y \in X \end{multline*} \]

which can be seen as a combination of both the midpoint and the trapezoid \(p\)-norm (with \(2\leq p<\infty\)) inequalities.

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