An inequality involving Bernstein polynomials and convex functions

Abel, Ulrich

doi:10.1016/j.jat.2017.05.006

“…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).…”

Section: (): V-volsupporting

confidence: 78%

On the Raşa Inequality for Higher Order Convex Functions

Komisarski

¹

,

Rajba

²

2021

Results Math

4

0

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We study the following $$(q-1)$$ ( q - 1 ) th convex ordering relation for qth convolution power of the difference of probability distributions $$\mu $$ μ and $$\nu $$ ν $$\begin{aligned} (\nu -\mu )^{*q}\ge _{(q-1)cx} 0 , \quad q\ge 2, \end{aligned}$$ ( ν - μ ) ∗ q ≥ ( q - 1 ) c x 0 , q ≥ 2 , and we obtain the theorem providing a useful sufficient condition for its verification. We apply this theorem for various families of probability distributions and we obtain several inequalities related to the classical interpolation operators. In particular, taking binomial distributions, we obtain a new, very short proof of the inequality given recently by Abel and Leviatan (2020).

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“…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]).…”

Section: Introductionmentioning

confidence: 99%

“…Following the ideas of [1], Gavrea [6] extended Theorem A. He considered more general functions a n,i (x) than p n,i (x) and, more importantly, he replaced the function evaluations…”

Section: Introductionmentioning

confidence: 99%

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

Abel

¹

,

Leviatan

²

2020

Self Cite

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We extend an inequality involving the Bernstein basis polynomials and convex functions on [0, 1]. The inequality was originally conjectured by Raşa about thirty years ago, but was proved only recently. Our extension provides an inequality involving q-monotone functions, $$q\in \mathbb N$$ q ∈ N . In particular, 1-monotone functions are nondecreasing functions, and 2-monotone functions are convex functions. In general, q-monotone functions on [0, 1], for $$q\ge 2$$ q ≥ 2 , possess a $$(q-2)$$ ( q - 2 ) nd derivative in (0, 1), which is convex there. We also discuss some other linear positive approximation processes.

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“…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].…”

Section: Introductionmentioning

confidence: 94%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Komisarski¹,

Rajba²

2018

Mathematical Inequalities &Amp; Applications

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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.

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An inequality involving Bernstein polynomials and convex functions

Cited by 15 publications

References 3 publications

On the Raşa Inequality for Higher Order Convex Functions

On the Raşa Inequality for Higher Order Convex Functions

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

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

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