Generalized Steffensen’s Inequality by Fink’s Identity

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. .

Generalized Taylor’s Formula and Steffensen’s Inequality

“…In fact, the results presented in [32] provide generalizations of all [10,21,31]. A few other variants of Steffensen's inequality by using interpolating polynomials can be seen in [33][34][35]. Moreover, to elaborate the importance of Hardy-type inequalities in the theory of function spaces, we recommend [36] to the readers.…”