A solution to the problem of Raşa connected with Bernstein polynomials

Abstract

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.

Convex order for convolution polynomials of Borel measures

“…Our investigation is motivated by the recent result of J. Mrowiec, T. Rajba and S. Wąsowicz [11] who proved the following convex ordering relation for convolutions of binomial distributions B(n, x) and B(n, y) (n ∈ N, x, y ∈ [0, 1]): which is a probabilistic version of the inequality involving Bernstein polynomials and convex functions, that was conjectured as an open problem by I. Raşa [12] (see also [1], [2], [8], [5] for further results on the I. Raşa problem).…”

Muirhead inequality for convex orders and a problem of I. Raşa on Bernstein polynomials

“…The Bernstein operator B n : C([0, 1]) → C([0, 1]) ( [3,4]) associates to each continuous function ϕ : [0, 1] → R the function B n (ϕ) : [0, 1] → R given by B n (ϕ)(x) = n i=0 b n,i (x)ϕ i n . Recently, J. Mrowiec, T. Rajba and S. Wąsowicz [12] proved the following theorem on inequality for Bernstein operators: [12] showed that the conjecture is true. Their proof makes heavy use of probability theory.…”

A sharpening of a problem on Bernstein polynomials and convex functions

“…Starting from these remarks, the second author presented the inequality (4) as an open problem in [10]. A probabilistic solution was found by A. Komisarski and T. Rajba [5] using the methods developed in [8] and [6].…”