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Mathematics 2019, 7(4), 329; https://doi.org/10.3390/math7040329

Generalized Steffensen’s Inequality by Fink’s Identity

1
Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan
2
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
3
RUDN University, Miklukho-Maklaya str. 6, 117198 Moscow, Russia
*
Author to whom correspondence should be addressed.
Received: 11 February 2019 / Revised: 27 March 2019 / Accepted: 28 March 2019 / Published: 4 April 2019
(This article belongs to the Special Issue Inequalities)
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PDF [316 KB, uploaded 22 April 2019]

Abstract

By using Fink’s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity, we give new generalizations of Steffensen’s inequality and its reverse. Generalizations of some inequalities (and their reverse), which are related to Hardy-type inequality. New bounds of Gr u ¨ ss and Ostrowski-type inequalities have been proved. Moreover, we formulate generalized Steffensen’s-type linear functionals and prove their monotonicity for the generalized class of ( n + 1 ) -convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions. View Full-Text
Keywords: Steffensen’s inequality; higher order convexity; Green functions; Montgomery identity; Fink’s identity Steffensen’s inequality; higher order convexity; Green functions; Montgomery identity; Fink’s identity
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Fahad, A.; Butt, S.I.; Pečarić, J. Generalized Steffensen’s Inequality by Fink’s Identity. Mathematics 2019, 7, 329.

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