Linear operators inequality for $n$-convex functions at a point

Abstract

Abstract. We study necessary and sufficient conditions on linear operators A and B for inequality A f B f to hold for every function f that is n -convex at a point.Mathematics subject classification (2010): 26D99, 26D15.

Generalization of cyclic refinements of Jensen’s inequality by Fink’s identity

“…In the present section we formulate related results for the class of n-convex functions at a point introduced by Pečarić et al in [12].…”

Popoviciu type inequalities for n-convex functions via extension of Montgomery identity

“…3 Related inequalities for n-convex functions at a point In this section we will give related results for the class of n-convex functions at a point introduced in [9]. Definition 3.…”

“…A property that explains the name of the class is the fact that a function is n-convex on an interval if and only if it is n-convex at every point of the interval (see [2,9]). Pečarić, Praljak and Witkowski in [9] studied necessary and sufficient conditions on two linear functionals A : C([a, c]) → R and B : C([c, b]) → R so that the inequality A(f ) ≤ B(f ) holds for every function f that is n-convex at c. In this section we will give inequalities of this type for particular linear functionals related to the inequalities obtained in the previous section.…”

“…In Section 3 we will give related inequalities for n-convex functions at a point, a generalization of the class of n-convex functions introduced in [9]. In Section 4 we will will give bounds for the integral reminders of identities obtained in earlier sections by usingČebyšev type inequalities.…”

“…The outline of the paper is as follows: in Section 2 we will use the extension of Montgomery's identity (9) to obtain inequalities of type (1) and (7) for n-convex functions. In Section 3 we will give related inequalities for n-convex functions at a point, a generalization of the class of n-convex functions introduced in [9].…”