Generalization of Levinson's inequality

Abstract

Abstract. Mercer [5] gave a generalization of Levinson's inequality that replaces the assumption of symmetry of the two sequences with a weaker assumptions of equality of variances. Witkowski [10] further loosened this assumption and extended the result to the class of 3-convex functions.We generalize these results to a newly defined, larger class of functions. We also prove the converse in case the function is continuous. In particular, we show that if Levinson's inequality holds under Mercer's assumptions, then the function is 3-convex.Mathematics subject classification (2010): 26D15.

Generalized Levinson's inequality and exponential convexity

“…It is obvious from the proof that Levinson's inequality (1.5) holds if the equality (2.1) is replaced by the weaker condition [3]), if, additionally, f is convex (resp. concave), this condition can be further weakened to Var(Y ) − Var(X) ≥ 0 (resp.…”

“…Baloch, Pečarić and Praljak ( [3]) also proved the converse of Levinson's inequality for continuous functions.…”

“…Building on Witkowski's ideas ( [12]), Baloch, Pečarić and Praljak ( [3]) showed that, in this case, Levinson's inequality holds for a larger class of functions given in the following definition. • is its interior.…”

“…We will also prove a converse stronger than Theorem 1.6 (ii). In addition to being more general, the proofs of the results from Section 2 will be significantly more elegant and intuitive than the rather technical and convoluted proofs from [3]. In Section 3 we will give mean value type results.…”

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

“…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.…”