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

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∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in \mathbb N$$\end{document}. 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≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ge 2$$\end{document}, possess a (q-2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q-2)$$\end{document}nd derivative in (0, 1), which is convex there. We also discuss some other linear positive approximation processes.


Introduction
Given a function f defined on [0, 1], the classical Bernstein polynomials associated with it are defined by, For the sake of simplicity we need also the notation p n,j (x) = 0, j < 0 and j > n.
one is convex. It is well known (see, e.g., [10]) that the Bernstein polynomials preserve q-monotonicity of all orders q ≥ 1. In view of the above, it is, thus, natural to ask whether there is an analog of Theorem A for any other q. The purpose of this paper is to present this analogous result. Namely, we will show that Theorem 1.1. Let q, n ∈ N. If f ∈ C [0, 1] is a q-monotone function, then for all x, y ∈ [0, 1], Here and in the sequel we follow the convention that an empty product is equal to 1.
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.
Remark 1.2. Note that for q = 1, inequality (1.1) is simply rewriting the fact that if f is nondecreasing, so are the Bernstein polynomials associated with it. We don't know if for any of the other q's, (1.1) is equivalent to the qmonotonicity preservation of the Bernstein polynomials.
Actually Theorem 1.1 will follow as a special case from Theorem 1.3 which we state after introducing some notation. With Our result is We will prove Theorem 1.3 in Sect. 2. Then, in Sect. 3, we will discuss the analogues of Theorem 1.3 for the basis elements of the Favard-Mirakyan-Szász and of the Baskakov operators.

Proof of Theorem 1.3
We begin with some auxiliary formulas.
where the last equality follows by Leibniz rule for the differentiation of products of functions. Note that the formula is valid also if m > n as we recall that p n,j (x) = 0, for j > n. Hence, for x, y ∈ [0, 1] and any (qn Vol. 75 (2020) An Extension of Raşa's Conjecture to q-Monotone Functions Page 5 of 13 180 Now, define a polynomial in z of degree at most (n − 1)q. Then, The next proposition is the key result.

Other Classical Linear Positive Approximation Operators
Given a function f defined on [0, ∞), we are going to discuss two classical linear approximation processes, the Favard-Mirakyan-Szász operators and the Baskakov operators associated with it. The Favard-Mirakyan-Szász operators, associated with f defined on [0, ∞), such that |f (x)| ≤ Ce Ax , x ∈ [0, ∞), for some constants C, A > 0, are defined by, If we denote then the operators may be represented as, We may generalize the operators as we have done in Sect. 1 to include both the above operators (for α = 0) and their Kantorovich polynomials' variant (for α = 1). Namely, for f as above which is integrable in every compact subinterval of [0, ∞), and α ∈ [0, 1], let It follows that, for a continuous f as above, If we denote k= 0, 1, . . . , (3.2) then we may write We generalize these operators in the above spirit, namely, for an appropriate f integrable in every compact subinterval of [0, ∞), and α ∈ [0, 1], let We have the following two results analogous to Theorem 1.3.
Proof of Theorem 3.2. Let n, m, q ∈ N, and denote ν : Hence, as is done in (2.2), for 0 ≤ x, y < ∞ and any sequence (a k ) ∞ k=0 , we have

Concluding Remarks
In recent years there has been a unified approach that includes both the Bernstein and Baskakov operators, yet, one has to keep in mind that the functions with which the operators are associated are defined on different intervals. For c = 0, denote