Remarks on q-Monotone Functions and the Bernstein polynomials

Mathematical Inequalities & Applications

In this paper we show that α -Bernstein operators preserve q -monotonicity of all orders. We investigate the tensor product of two such operators and show that it preserves (q,s)box convexity. Some Ras ¸a type inequalities for the α -Bernstein operators are also derived. Mathematics subject classification (2020): 41A36, 41A17.

Applied Mathematics-A Journal of Chinese Universities, 2006

In this paper a class of new inequalities about Bernstein polynomial is established.

arXiv (Cornell University), 2018

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,

We characterize of the q-Bernstein functions in terms of q-Laplace transform. Moreover, we present several results of q-completely monotonic, q-log completely monotonic and q-Bernstein functions.

2015

We characterize of the q-Bernstein functions in terms of q-Laplace transform. Moreover, we present several results of q-completely monotonic, q-log completely monotonic and q-Bernstein functions.

Mediterranean Journal of Mathematics

We prove that various Durrmeyer-type operators preserve q-monotonicity in [0, 1] or $$[0,\infty )$$ [ 0 , ∞ ) as the case may be. Recall that a 1-monotone function is nondecreasing, a 2-monotone one is convex, and for $$q>2$$ q > 2 , a q-monotone function possesses a convex $$(q-2)$$ ( q - 2 ) nd derivative in the interior of the interval. The operators are the Durrmeyer versions of Bernstein (including genuine Bernstein–Durrmeyer), Szász and Baskakov operators. As a byproduct we have a new type of characterization of continuous q-monotone functions by the behavior of the integrals of the function with respect to measures that are related to the fundamental polynomials of the operators.

Applied Mathematics & Information Sciences, 2017

The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified q-Bernoulli numbers and q-Euler numbers applying p-adic q-integral representation on Z p and p-adic fermionic q-invariant integral on Z p , respectively, to the inverse of q-Bernstein polynomials.

Journal of Mathematical Analysis and Applications, 2009

In this paper, we introduce the generalized q-Bernstein polynomials based on the q-integers and we study approximation properties of these operators. In special case, we obtain Stancu operators or Phillips polynomials.

arXiv: Probability, 2015

We give several new characterizations of completely monotone functions and Bernstein functions via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on the set of non-negative integers. We give a complete answer to the following question: Can we affirm that a function is completely monotone (resp. a Bernstein function) if we know that the sequence formed by its restriction on the integers is completely monotone (resp. alternating)? This approach constitutes a kind of converse of Hausdorff's moment characterization theorem in the context of completely monotone sequences.

Communications in Mathematics and Applications

This paper deals with a sequence of the combination of Bernstein polynomials with a positive function τ and based on a parameter s > − 1 2. These polynomials have preserved the functions 1 and τ. First, the convergence theorem for this sequence is studied for a function f ∈ C[0, 1]. Next, the rate of convergence theorem for these polynomials is descript by using the first, second modulus of continuous and Ditzian-Totik modulus of smoothness. Also, the Quantitative Voronovskaja and Grüss-Voronovskaja are obtained. Finally, two numerical examples are given for these polynomials by chosen a test function f ∈ C[0, 1] and two functions for τ to show that the effect of the different values of s and the different chosen functions τ.