Generalized Steffensen’s inequality by Montgomery identity

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 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. MSC: Primary 26D10; secondary 26D20Keywords: Steffensen's inequality; Green's function; Montgomery's identity; (n + 1)-convex function at a point c+θ c ψ(z) dz, where θ = d c f (z) dz.( 1 )A massive literature body dealing with several variants and improvements of (1) can be seen in [14,16] and the references therein. Pečarić [13] gave a nice generalization of (1) as follows.

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.…”

Extension of Montgomery identity via Taylor polynomial on time scales

“…The interpolation (1.1) is used by many authors to generalized inequalities for higher order convex function, namely Jensen's inequaity [12], Jensen-Stefensen's inequality [13], Sherman's inequality [14,15], cyclic refinement of Jensen's inequality [8], Popoviciu's inequality [9,16], combinatorial improvements of Jensen's inequality [18], Levinson's inequality [1] via time scales theory [2].…”