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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. [ABSTRACT FROM AUTHOR]

We present some fundamental results and definitions regarding Jensen's inequality with the aim of obtaining new generalizations of cyclic refinements of Jensen's inequality from convex to higher order convex functions using Taylor's formula. We discuss the monotonicity of functionals for n-convex functions at a point. Applications of our work include new bounds for some important inequalities used in information theory. [ABSTRACT FROM AUTHOR]

We generalize cyclic refinements of Jensen’s inequality from a convex function to a higher-order convex function by means of Lagrange–Green’s function and Fink’s identity. We formulate the monotonicity of the linear functionals obtained from these identities utilizing the theory of inequalities for n-convex functions at a point. New Grüss- and Ostrowski-type bounds are found for identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity and mean value theorems. [ABSTRACT FROM AUTHOR]