Generalized Steffensen’s Inequality by Fink’s Identity

• Main Authors: Asfand Fahad, Saad Ihsan Butt, Josip Pečarić
• Format: Article
• Language: English
• Published: MDPI AG 2019-04-01
• Series: Mathematics
• Subjects: Steffensen’s inequality; higher order convexity; Green functions; Montgomery identity; Fink’s identity
• Online Access: https://www.mdpi.com/2227-7390/7/4/329

By using Fink&#8217;s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen&#8217;s inequality. Under the assumptions of <i>n</i>-convexity and <i>n</i>-concavity, we give new generalizations of Steffensen&#8217;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>&uml;</mo> </mover> </semantics> </math> </inline-formula>ss and Ostrowski-type inequalities have been proved. Moreover, we formulate generalized Steffensen&#8217;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.