Jensen–Steffensen inequality for diamond integrals, its converse and improvements via Green function and Taylor’s formula

Nosheen, Ammara; Bibi, Rabia; Pečarić, Josip

doi:10.1007/s00010-017-0527-2

Cited by 7 publications

(2 citation statements)

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“…By using these identities we obtain a generalization of (4). In addition, we construct new identities which enable us to prove generalizations of inequalities (5) and (6) as one can obtain Classical Hardy-type inequalities from them, see [1]. We useČebyšev functional to construct new bounds of Grüss and Ostrowski-type inequalities.…”

Section: Theoremmentioning

confidence: 99%

“…Several generalizations of these inequalities have been proved for different classes of functions, such as convex functions, n-convex functions, and other types of functions, for example see [1][2][3][4]. Moreover, integral inequalities have been proved for different integrals, such as Jensen-steffensen inequality for diamond integral and bounds of related identities have been obtained in [5]. Other than that, Hardy's inequality for fractional integral on general domains have been proved in [6].…”

Section: Introductionmentioning

confidence: 99%

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Generalized Steffensen’s Inequality by Fink’s Identity

2019

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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. .

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Section: Theoremmentioning

confidence: 99%