During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Ioan Raşa recalled his 25-years-old problem concerning some inequality involving the Bernstein polynomials. We offer the complete solution (in positive). As a tool we use stochastic orderings (which we prove for binomial distributions) as well as so-called concentration inequality. Our methods allow us to pose (and solve) the extended version of the problem in question.
Recently Nikodem, Rajba and Wąsowicz compared the classes of n-Wrightconvex functions and n-Jensen-convex functions by showing that the first one is a proper subclass of the latter one, whenever n is an odd natural number. Till now the case of even n was an open problem. In this paper the complete solution is given: it is shown that the inclusion is proper for any natural n. The classes of strongly n-Wright-convex and strongly n-Jensen-convex functions are also compared (with the same assertion).
Keywords:Convex functions of higher order Jensen-convex functions of higher order Wright-convex functions of higher order Forward difference Backward difference Hamel basis Dirac measure a b s t r a c tThe classes of n-Wright-convex functions and n-Jensen-convex functions are compared with each other. It is shown that for any odd natural number n the first one is the proper subclass of the second one. To reach this aim new tools connected with measure theory are developed.
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