Rad HAZU, Matematičke znanosti, Vol. 19 (2015), 69-89.

WEIGHTED POPOVICIU TYPE INEQUALITIES VIA GENERALIZED MONTGOMERY IDENTITIES

Saad Ihsan Butt and Josip Pečarić

Department of Mathematics, Faculty of Textile Technology, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: pecaric@mahazu.hazu.hr

Abstract.   We obtained useful identities via generalized Montgomery identities, by which the inequality of Popoviciu for convex functions is generalized for higher order convex functions. We investigate the bounds for the identities related to the generalization of the Popoviciu inequality using inequalities for the Čebyšev functional. Some results relating to the Grüss and Ostrowski type inequalities are constructed. Further, we also construct new families of exponentially convex functions and Cauchy-type means by looking at linear functionals associated with the obtained inequalities.

2010 Mathematics Subject Classification.   26D07, 26D15, 26D20, 26D99.

Key words and phrases.   Convex function, divided difference, generalized Montgomery identity, Čebyšev functional, Grüss inequality, Ostrowski inequality, exponential convexity.

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