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The inequality of Popoviciu, which was improved by Vasić and Stanković (Math. Balk. 6:281-288,  1976 ), is generalized by using new identities involving new Green’s functions. New generalizations of an improved Popoviciu inequality are obtained by using generalized Montgomery identity along with new Green’s functions. As an application, we formulate the monotonicity of linear functionals constructed from the generalized identities, utilizing the recent theory of inequalities for n-convex functions at a point. New upper bounds of Grüss and Ostrowski type are also computed.

New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity