Shape-preserving properties of a new family of generalized Bernstein operators

In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\alpha,q)$\end{document}(α,q)-Bernstein operators, denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{n,q,\alpha}(f)$\end{document}Tn,q,α(f). We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{n,q,\alpha}(f)$\end{document}Tn,q,α(f) to any continuous functions f. The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x)$\end{document}f(x). We also obtain the monotonicity with n and q of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{n,q,\alpha}(f)$\end{document}Tn,q,α(f).


Introduction
A generalization of Bernstein polynomials based on q-integers was proposed by Lupaş in 1987 in [1]. However, the Lupaş q-Bernstein operators are rational functions rather than polynomials. In 1997, Phillips [2] proposed the Phillips q-Bernstein polynomials, and for decades thereafter the application of q integers in positive linear operators became a hot topic in approximation theory, such as generalized q-Bernstein polynomials [3][4][5][6], Durrmeyer-type q-Bernstein operators [7][8][9], Kantorovich-type q-Bernstein operators [10][11][12][13], etc. As we know, q integers play important roles not only in approximation theory, but also in CAGD. Based on the Phillips q-Bernstein polynomials [2], which are generalizations of Bernstein polynomials, generalized Bézier curves and surfaces were introduced in [14][15][16]. In [14], Oruç and Phillips constructed q-Bézier curves using the basis functions of Phillips q-Bernstein polynomials. Dişibüyük and Oruç [15,16] defined the q generalization of rational Bernstein-Bézier curves and tensor product q-Bernstein-Bézier surfaces. Moreover, Simeonov et al. [17] introduced a new variant of the blossom, the q blossom, which is specifically adapted to developing identities and algorithms for q-Bernstein bases and q-Bézier curves. In 2014, Han et al. [18] proposed a generalization of q-analog Bézier curves with one shape parameter, and established degree evaluation and de Casteljau algorithms and some other properties. In 2016, Han et al. [19] introduced a new generalization of weighted rational Bernstein-Bézier curves based on q integers, and investigated the generalized rational Bézier curve from a geometric point of view, obtaining degree evaluation and de Casteljau algorithms, etc.
Recently, Chen et al. [20] introduced a new family of α-Bernstein operators, and investigated some approximation properties, such as the rate of convergence, Voronovskaja-type asymptotic formulas, etc. They also obtained the monotonic and convex properties. For f (x) ∈ [0, 1], n ∈ N, and any fixed real α, the α-Bernstein operators they introduced are defined as where where n ≥ 2. Motivated by above research, in this paper we propose the q analogue of α-Bernstein operators, called (α, q)-Bernstein operators, which are defined as where q ∈ (0, 1], [n] q ), i = 0, 1, 2, . . . , n, p (α) 1,q,0 (x) = 1x, p (α) 1,q,1 (x) = x, and By simple computations, we can also express the (α, q) operators (3) as where Here, we mention some definitions based on q integers, the details of which can be found in [21,22]. For any fixed real number 0 < q ≤ 1 and each non-negative integer k, we denote Also, q-factorial and q-binomial coefficients are defined as follows: The q-analog of (1 + x) n is defined by (1 + x) n q := n-1 s=0 (1 + q s x). The q derivative and q derivative of the product are defined as D q f (x) := q . The rest of this paper is organized as follows. In the next section, we give some basic properties of the operators T n,q,α (f ), such as the moments and central moments for proving the convergence theorems, the forward difference form of T n,q,α (f ) for proving shape-preserving properties, etc. In Sect. 3, we obtain the convergence property and the rate of convergence theorem. In Sect. 4, we investigate some shape-preserving properties, such as monotonicity-and convexity-preserving properties with respect to f (x), and also we study the monotonicity with n and q of T n,q,α (f ).

Auxiliary results
For proving the main results, we require the following lemmas.

Lemma 2.1
We have the following equalities: Proof By (5), we have However, Lemma 2.1 is proved.
Remark 2.2 From Lemma 2.1, we know that the (α, q)-Bernstein operators T n,q,α (f ; x) reproduce linear functions; that is, for all real numbers a and b.

Lemma 2.4
(i) The (α, q)-Bernstein operators may be expressed in the form where The higher-order forward difference of g i may be expressed in the form where 0 q g i = g i , which is defined in (6).
Proof We can obtain (8) easily by [2]. Next, in order to prove (9), we use induction on r. It is clear that (9) holds for r = 0. Let us assume that (9) holds for some r = k ≥ 0. For r = k + 1, we have This shows that (9) holds when k is replaced by k + 1, and this completes the proof of Lemma 2.4.
[n] q ), the q differences of the monomial x k of order greater than k are zero. We see from Lemma 2.4 that, for all n ≥ k, T n,q,α (t k ; x) is a polynomial of degree k. Actually, the (α, q)-Bernstein operators are degree-reducing on polynomials; that is, if f is a polynomial of degree m, and then T n,q,α (f ) is a polynomial of degree ≤ min{m, n}. In particular, we have the following results.
Proof Indeed, from (9) and Thus, we obtain k q g 0 = 1 + Hence, using (8), we have We then obtain the proof of Lemma 2.5 by simple computations.
Then (11) is proved by (10). This completes the proof of Lemma 2.6.

Convergence properties
We now state the well-known Bohman-Korovkin theorem, followed by a proof based on that given by Cheney [23]. Theorem 3.1 leads to the following theorem on the convergence of (α, q)-Bernstein operators. Proof From Lemma 2.1, we see that T n,q,α (f ; x) = f (x) for f (t) = 1 and f (t) = t. Since lim n→∞ q n = 1, we see from (10) that T n,q,α (f ; x) converges uniformly to f (x) for f (t) = t 2 as n → ∞. It also follows that T n,q,α is a monotone operator by Lemma 2.3; the proof is then completed by applying the Bohman-Korovkin theorem 3.1. [24], there exists an absolute constant C > 0, such that
where C is a positive constant.
Remark 3.4 Letting q := {q n } denote a sequence such that q n ∈ (0, 1) and lim n→∞ q n = 1, we know that, under the conditions of theorem 3.3, the convergence rate of the operators T n,q,α (f ) to f is 1/ [n] q as n → ∞. This convergence rate can be improved depending on the choice of q, at least as fast as 1/ √ n.
Example 3.5 Letting f (x) = 1 -cos(4e x ), the graphs of f (x) and T n,q,0.9 (f ; x) with different values of n and q are shown in Fig. 1. Figure 2 shows the graphs of f (x) and T 10,0.9,α (f ; x) with α = 0.6 and α = 0.9.

Figure 2
Convergence of T n,q,α (f ; x) to f (x) for fixed q = 0.9 [n] q f i+1 . Then the q derivative of T n+1,q,α (f ; x) is and we denote the first and second parts of the right-hand side of the last equation by 1 and 2 , respectively. We then have Using (9), we obtain 1 q g i = 1 - Thus, we have Similarly, we can obtain Therefore, by using (15) and (16), the derivative of (α, q)-Bernstein operators T n,q,α (f ; x) may be expressed in the form D q T n,q,α (f ; x) Since if f is monotonically increasing on [0, 1], the forward differences 1 q f i and 1 q f i+1 are non-negative, and so is D q [T n,q,α (f ; x)]. Hence, (α, q)-Bernstein operators T n,q,α (f ; x) are monotonically increasing on [0, 1] for fixed q ∈ (0, 1) and α ∈ [0, 1]. On the contrary, if f is monotonically decreasing on [0, 1], then operators T n,q,α (f ; x) are monotonically decreasing on [0, 1] for fixed q ∈ (0, 1) and α ∈ [0, 1]. Theorem 4.1 is proved.
The (α, q)-Bernstein operators T n,q,α (f ; x) have a convexity-preserving property Proof From (5), we obtain The q-derivative of T n+2,q,α (f ; x) can easily obtained by the proof theorem 4.1, which may be expressed as Then we have By some easy computations, we obtain By the connection between the secondorder q differences and convexity, we know that 2 q f i and 2 q f i+1 are all non-negative since f is convex on [0, 1]. Hence, we obtain D 2 q [T n+2,q,α (f ; x)] ≥ 0, and then the convexitypreserving property of T n,q,α (f ; x). Theorem 4.2 is proved.
Next, if f (x) is convex, the (α, q)-Bernstein operators T n,q,α (f ; x), for n fixed, are monotonic in q.
Proof In the following main proof of our results, we must introduce a linear polynomial function: where [n] q ), i = 0, . . . , n -1. Then it is straightforward to check that g i = g( [i] q [n-1] q ). Since f is convex on [0, 1], the intrinsic linear polynomial function g(x) must be convex on [0, 1] as well. Therefore, by the classical results of q-Bernstein operators (see [3]), we note that T n,q,α (f ; x) = (1α)B q n-1 (g; x) + αB q n (f ; x).
We have B Finally, if f (x) is convex, we give the monotonicity of (α, q)-Bernstein operators T n,q,α (f ; x) with n.

Conclusion
In this paper, we proposed a new family of generalized Bernstein operators, named (α, q)-Bernstein operators, and denoted by T n,q,α (f ). We study the rate of convergence of these operators, investigate their monotonicity-, convexity-preserving properties with respect to f (x), and also obtain their monotonicity with n and q of T n,q,α (f ).