Perturbed weighted trapezoid inequalities for convex functions with applications

Sever Silvestru Dragomir; Eder Kikianty

doi:10.56754/0719-0646.2603.507

DOI:

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

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.

Keywords

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

Mathematics Subject Classification:

26D15 , 46C50

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

Pages: 507–523
Date Published: 2024-12-12
Vol. 26 No. 3 (2024)

P. Cerone and S. S. Dragomir, “A refinement of the Grüss inequality and applications,” Tamkang J. Math., vol. 38, no. 1, pp. 37–49, 2007.

P. Chebyshev, “Sur les expressions approximatives des intégrales définies par les autres prises entre les mêmes limites,” Proc. Math. Soc. Charkov, vol. 2, pp. 93–98, 1882.

X.-L. Cheng and J. Sun, “A note on the perturbed trapezoid inequality,” JIPAM. J. Inequal. Pure Appl. Math., vol. 3, no. 2, 2002, Art. ID 29.

I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, ser. Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1990, vol. 62, doi: 10.1007/978-94-009-2121-4.

S. S. Dragomir, “An inequality improving the first Hermite-Hadamard inequality for con- vex functions defined on linear spaces and applications for semi-inner products,” JIPAM. J. Inequal. Pure Appl. Math., vol. 3, no. 2, 2002, Art. ID 31.

S. S. Dragomir, “An inequality improving the second Hermite-Hadamard inequality for con- vex functions defined on linear spaces and applications for semi-inner products,” JIPAM. J. Inequal. Pure Appl. Math., vol. 3, no. 3, 2002, Art. ID 35.

S. S. Dragomir, “Smooth normed spaces of (BD)-type,” J. Fac. Sci. Univ. Tokyo Sect. IA Math., vol. 39, no. 1, pp. 1–15, 1992.

S. S. Dragomir, Semi-inner products and applications. Nova Science Publishers, Inc., Haup-pauge, NY, 2004.

S. S. Dragomir, “Ostrowski type inequalities for Lebesgue integral: a survey of recent results,” Aust. J. Math. Anal. Appl., vol. 14, no. 1, 2017, Art. ID 1.

L. Fejér, “Über die Fouriersche Reihe, II,” Mat. Természett. Értes., vol. 24, pp. 369–390, 1906.

J. R. Giles, “Classes of semi-inner-product spaces,” Trans. Amer. Math. Soc., vol. 129, pp. 436–446, 1967, doi: 10.2307/1994599.

G. Grüss, “Über das Maximum des absoluten Betrages von {(frac{1}{{b - a}}intlimits_a^b {fleft( x right)} gleft( x right)dx - frac{1}{{left( {b - a} right)^2 }}intlimits_a^b {fleft( x right)dx} intlimits_a^b g left( x right)dx)}}” Math. Z., vol. 39, no. 1, pp. 1935, doi: 10.1007/BF01201355.

E. Kikianty, S. S. Dragomir, and P. Cerone, “Sharp inequalities of Ostrowski type for convex functions defined on linear spaces and application,” Comput. Math. Appl., vol. 56, no. 9, pp. 2235–2246, 2008, doi: 10.1016/j.camwa.2008.03.059.

E. Kikianty, S. S. Dragomir, and P. Cerone, “Ostrowski type inequality for absolutely continuous functions on segments in linear spaces,” Bull. Korean Math. Soc., vol. 45, no. 4, pp. 763–780, 2008, doi: 10.4134/BKMS.2008.45.4.763.

G. Lumer, “Semi-inner-product spaces,” Trans. Amer. Math. Soc., vol. 100, pp. 29–43, 1961, doi: 10.2307/1993352.

A. M. Ostrowski, “On an integral inequality,” Aequationes Math., vol. 4, pp. 358–373, 1970, doi: 10.1007/BF01844168.