| Summary: | By using Fink’s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen’s inequality. Under the assumptions of <i>n</i>-convexity and <i>n</i>-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<inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>u</mi> <mo>¨</mo> </mover> </semantics> </math> </inline-formula>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 <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-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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