Generalized Steffensen’s inequality by Montgomery identity

Abstract

Abstract By using generalized Montgomery identity and Green functions we proved several identities which assist in developing connections with Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity many inequalities, which generalize Steffensen’s inequality, inequalities from (Fahad et al. in J. Math. Inequal. 9:481–487, 2015; Pečarić in Southeast Asian Bull. Math. 13:89–91, 1989; Rabier in Proc. Am. Math. Soc. 140:665–675, 2012), and their reverse, have been proved. Generalization of some inequalities (and their reverse) which are related to Hardy-type inequality (Fahad et al. in J. Math. Inequal. 9:481–487, 2015) have also been proved. New bounds of Ostrowski and Grüss type inequalities have been developed. Moreover, we formulate generalized Steffensen-type linear functionals and prove their monotonicity for the generalized class of (n+1) $(n+1)$-convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions.

Keywords

• Steffensen’s inequality
• Green’s function
• Montgomery’s identity
• ( n + 1 ) $(n+1)$ -convex function at a point