A sharpening of a problem on Bernstein polynomials and convex functions and related results

Abstract

We present a short proof of a conjecture proposed by I. Raşa (2017), which is an inequality involving basic Bernstein polynomials and convex functions. This proof was given in the letter to I. Raşa (2017). The methods of our proof allow us to obtain some extended versions of this inequality as well as other inequalities given by I. Raşa. As a tool we use stochastic convex ordering relations. We propose also some generalizations of the binomial convex concentration inequality. We use it to insert some additional expressions between left and right sides of the Raşa inequalities.is valid for all x, y ∈ [0, 1].This inequality involving Bernstein basic polynomials and convex functions was stated as an open problems 25 years ago by I. Raşa. During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Raşa [12] recalled his problem.Inequalities of type (1.1) have important applications. They are useful when studying whether the Bernstein-Schnabl operators preserve convexity (see [3,4]).Recently, J. Mrowiec, T. Rajba and S. Wąsowicz [10] affirmed the conjecture (1.1) in positive. Their proof makes heavy use of probability theory. As a tool they applied a concept of stochastic convex orderings, as well as the so-called binomial convex concentration inequality. Later, U. Abel [1] gave an elementary proof of (1.1), which was much shorter than that given in [10]. Very recently, A. Komisarski and T. Rajba [6] gave a new, very short proof of (1.1), which is significantly simpler and shorter than that given by U. Abel [1]. As a tool the authors use both stochastic convex orders as well as the usual stochastic order.Let us recall some basic notations and results on stochastic ordering (see [14]). If µ and ν are two probability distributions such that ϕ(x)µ(dx) ≤ ϕ(x)ν(dx) for all convex functions ϕ : R → R, 2010 Mathematics Subject Classification. 60E15, 39B62.

Convex order for convolution polynomials of Borel measures

“…then the inequality (4.18) coincides with (4.19), which was proved in [8]. (a n,i (x) a n,j (x) + a n,i (y) a n,j (y) − 2a n,i (x) a n,j (y)) ϕ i + j 2n + i + j 0.…”

“…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).…”

On the Raşa Inequality for Higher Order Convex Functions

“…During the Conference on Ulam's Type Stability (Rytro, Poland, 2014), Raşa [14] recalled his problem. Theorem 1 affirms the conjecture (see also [1][2][3][4]7,8,15] for further results on the I. Raşa problem).…”

An Extension of Raşa’s Conjecture to q-Monotone Functions

“…As a tool they applied stochastic convex orderings (which they proved for binomial distributions) as well as the so-called concentration inequality. Recently [1], the first author gave an elementary proof of Theorem A, which has brought Komisarski and Rajba [8] to give a simple proof unifying that of Theorem A and of similar inequalities for the Favard-Mirakyan-Szász and Baskakov operators (see below), using probabilistic tools and the Hardy-Littlewood-Polya inequality (see also [9]).…”

“…Regrettably, the tools of stochastic processes and analysis that have been so elegantly used in [8,9] for the convex case, are not available for q > 2.…”