In this paper, we use alpha-Bernstein operators to solve an open problem related to box-convex functions. Future targeted applications reside in the area of stochastic optimization in problems such as AUC maximization and stochastic programs with chance constraints.

In this paper we study local approximation properties of a higher order Kantorovich-type Szasz-Mirakjan operator recently introduced by Sabancigil, Kara, and Mahmudov. We derive the complete asymptotic expansion for these operators. They generalize the Szasz-Mirakjan operators of Kantorovich-type and approximate locally integrable functions satisfying a certain growth condition on the infinite interval (0, infinity).

We provide a decomposition formula for the classical polynomial interpolation operator and obtain the generalized Hermite interpolant through a limiting process. As a consequence of our results, we obtain/reobtain known and new identities related to interpolation theory.

The authors provide a complete asymptotic expansion for a class of functions in terms of the complete Bell polynomials. In particular, they obtain known asymptotic expansions of some Keller type sequences.

We provide a mean-value theorem for a class of positive linear functionals. As an application, we improve the classical First Mean-value Theorem for Integrals and obtain other related results.

We answer a recent question of Bustamante concerning a representation formula for Bernstein operators.

We prove that the second-order moment of the Meyer–König and Zeller operators is an elementary function and find sharp forms of the related Becker and Nessel inequalities.

We prove that a property concerning the existence of the fixed points of the Bernstein-type operators remains valid for a large class of positive linear operators. More exactly, we prove that there is no positive linear analytic operator L : C[0, 1] -> C[0, 1] possessing as fixed points two nonconstant monomials x(i). (C) 2017 Elsevier Inc. All rights reserved.

We answer and generalize an open problem on the two-dimensional Bernstein polynomials on the unit triangle.

Our purpose is to study the local rate of convergence of the Gamma operators. The talk presents the complete asymptotic expansion for the Gamma operators. We investigate their asymptotic behavior also concerning simultaneous approximation. All expansion coefficients are explicitly calculated. It turns out that Stirling numbers play an important role. Moreover, we deal with linear combinations of Gamma operators having a better degree of approximation than the operators themselves. Using divided differences we define general classes of linear combinations, special cases of which were recently introduced and investigated by other authors. Finally, we study the left quasi-interpolants of the Gamma operators in the sense of Sablonnière.

We establish some mean value theorems involving n-simple functionals in the sense of Popoviciu. In particular, we obtain the Kowalewski mean value formula.

Abstract We answer a conjecture and an open problem concerning integral sequences of the form ∫ 0 1 f ⁢ ( x ) ⁢ f ⁢ ( x 2 ) ⁢ ⋯ ⁢ f ⁢ ( x n ) ⁢ d x n , n ≥ 2 . \sqrt[n]{\int_{0}^{1}f(x)f(x^{2})\cdots f(x^{n})\,\mathrm{d}x},\quad n\geq 2.

We define a new sequence of positive linear approximation operators by means of the squared Bernstein polynomials and estimate the rate of approximation.

We provide complete asymptotic expansions for some sequences of Bernstein-type and Meyer–König and Zeller-type operators preserving the monomials e0 and ej, j>1. In particular, this answers a conjecture related to a Voronovskaja-type theorem.

We provide an analysis of the rate of convergence of positive linear approximation operators defined on C[0, 1]. We obtain a sufficient condition for a sequence of positive linear approximation operators to possess a Mamedov-type property and give an application to the Durrmeyer approximation process.

We extend and solve an open problem concerning a Pólya–Szegő type integral sequence of the form \({\sqrt[n]{\int\nolimits_{0}^{1} \prod\nolimits_{k=1}^n f(x^{k^\alpha}) \mathrm{d}x}}\), \({n\ge2}\), \({\alpha\ge 0}\).

Recently, Chang and Xu gave a probabilistic proof of a combinatorial identity which involves binomial coefficients. Duarte and Guedes de Oliveira (J Integer Seq 16, 2013) extended the result. Applying a generalization of the Leibniz rule for higher derivatives of the product of functions yields a new short proof and a generalization of the above mentioned identity.

In 1984 D.D. Stancu defined a sequence of positive linear operators generalizing the Bernstein polynomials. In this paper a Durrmeyer variant of these operators is considered. We derive some basic approximation properties and present a complete asymptotic expansion for the sequence of these operators. All coefficients are calculated in a concise form.

We prove that the classical Bernstein Voronovskaja-type theorem remains valid in general for all sequences of positive linear approximation operators.