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) -convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions. © 2019, The Author(s).