Generalized Levinson's inequality and exponential convexity

Abstract

Abstract. We give a probabilistic version of Levinson's inequality under Mercer's assumption of equal variances for the family of 3-convex functions at a point. We also show that this is the largest family of continuous functions for which the inequality holds. New families of exponentially convex functions and related results are derived from the obtained inequality.

Levinson type inequalities for higher order convex functions via Abel–Gontscharoff interpolation

“…In [8], Adeel et al generalized Levinson's inequality for 3-convex function by using two Green functions. In [9], Pečarić et al gave a probabilistic version of Levinson's inequality (2) under Mercer's assumption of equal variances (but for a different number of data points) for the family of 3-convex functions at a point. They showed that this is the largest family of continuous functions for which inequality (2) holds.…”

Superadditivity of the Levinson functional and applications

“…In fact, it corresponds to a probabilistic version of the Levinson inequality derived in [10] (see Theorem 2.3). The proof in our setting is given for the reader's convenience.…”

“…Motivated by Witkowski's ideas, Pečarić et al [10] (see also [1]), showed that under Mercer's assumption (1.4) the Levinson inequality holds for a more general class of functions described in the following definition. Definition 1.1 Let : I → R and c ∈ I 0 , where I 0 is the interior of the interval I .…”

“…As shown in [10], K c 1 (I ) is the largest class of functions for which the Levinson inequality holds. In other words, the class K c 1 (I ) characterizes the Levinson inequality.…”

“…Pečarić et al [10], proved a more general probabilistic version of the Levinson inequality under the assumption of equality of variances. In particular, they showed that in the discrete Levinson inequality the number of points of the two sequences and associated weights do not need to be the same.…”