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

Abstract

In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called $(\alpha,q)$-Bernstein operators, denoted by $T_{n,q,\alpha}(f)$. We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of $T_{n,q,\alpha}(f)$ 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 $f(x)$. We also obtain the monotonicity with n and q of $T_{n,q,\alpha}(f)$.

1 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 [3–6], Durrmeyer-type q-Bernstein operators [7–9], Kantorovich-type q-Bernstein operators [10–13], 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 [14–16]. 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 $f(x)\in [0,1]$, $n\in \mathbb{N}$, and any fixed real α, the α-Bernstein operators they introduced are defined as

$$\begin{aligned} T_{n,\alpha }=\sum_{i=0}^{n}f_{i}p_{n,i}^{(\alpha )}(x), \end{aligned}$$

(1)

where $f_{i}=f ( \frac{i}{n} )$. For $i=0,1,\ldots,n$, the α-Bernstein polynomial $p_{n,i}^{\alpha }(x)$ of degree n is defined by $p_{1,0}^{(\alpha )}(x)=1-x$, $p_{1,1}^{(\alpha )}(x)=x$ and

$$\begin{array}{rcl}{p}_{n,i}^{(\alpha )}(x)& =& [\left(\begin{array}{c}n-2\\ i\end{array}\right)(1-\alpha )x+\left(\begin{array}{c}n-2\\ i-2\end{array}\right)(1-\alpha )(1-x)+\left(\begin{array}{c}n\\ i\end{array}\right)\alpha x(1-x)]\\ & & \times x^{i-1}{(1-x)}^{n-1-i},\end{array}$$

(2)

where $n\geq 2$.

Motivated by above research, in this paper we propose the q analogue of α-Bernstein operators, called $(\alpha , q)$-Bernstein operators, which are defined as

$$ T_{n,q,\alpha }(f;x)=\sum_{i=0}^{n}f_{i}p_{n,q,i}^{(\alpha )}(x), $$

(3)

where $q\in (0,1]$, $f_{i}=f ( \frac{[i]_{q}}{[n]_{q}} )$, $i=0,1,2,\ldots,n$, $p_{1,q,0}^{(\alpha )}(x)=1-x$, $p_{1,q,1}^{(\alpha )}(x)=x$, and

$$\begin{array}{rcl}{p}_{n,q,i}^{(\alpha )}(x)& =& ({\left[\begin{array}{c}n-2\\ i\end{array}\right]}_q(1-\alpha )x+{\left[\begin{array}{c}n-2\\ i-2\end{array}\right]}_q(1-\alpha )q^{n-i-2}(1-q^{n-i-1}x)\\ & & +{\left[\begin{array}{c}n\\ i\end{array}\right]}_q\alpha x(1-q^{n-i-1}x))x^{i-1}{(1-x)}_q^{n-i-1}(n\ge 2).\end{array}$$

(4)

By simple computations, we can also express the $(\alpha , q)$ operators (3) as

$$\begin{array}{c}{T}_{n,q,\alpha }(f;x)\hfill \\ =(1-\alpha )\sum _{i=0}^{n-1}{g}_i{\left[\begin{array}{c}n-1\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-1-i}+\alpha \sum _{i=0}^n{f}_i{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-i},\hfill \end{array}$$

(5)

where

$$\begin{aligned} g_{i}= \biggl( 1-\frac{q^{n-1-i}[i]_{q}}{[n-1]_{q}} \biggr) f_{i}+ \frac{q ^{n-1-i}[i]_{q}}{[n-1]_{q}}f_{i+1}. \end{aligned}$$

(6)

Here, we mention some definitions based on q integers, the details of which can be found in [21, 22]. For any fixed real number $0< q\leq 1$ and each non-negative integer k, we denote q-integers by $[k]_{q}$, where

$$ [k]_{q}:= \textstyle\begin{cases} \frac{1-q^{k}}{1-q}, &q\neq 1, \\ k, &q=1. \end{cases} $$

Also, q-factorial and q-binomial coefficients are defined as follows:

$$\begin{array}{c}{[k]}_q!:=\{\begin{array}{cc}{[k]}_q{[k-1]}_q\cdots {[1]}_q,\hfill & k=1,2,\dots ,\hfill \\ 1,\hfill & k=0,\hfill \end{array}\hfill \\ {\left[\begin{array}{c}n\\ k\end{array}\right]}_q:=\frac{{[n]}_q!}{{[k]}_q!{[n-k]}_q!}(n\ge k\ge 0).\hfill \end{array}$$

The q-analog of $(1+x)^{n}$ is defined by $(1+x)_{q}^{n}:=\prod_{s=0}^{n-1} ( 1+q^{s}x )$. The q derivative and q derivative of the product are defined as $D_{q}f(x):=\frac{d_{q}f(x)}{d _{q}x}=\frac{f(qx)-f(x)}{(q-1)x}$ and $D_{q}(f(x)g(x)):=f(qx)D_{q}g(x)+g(x)D _{q}f(x)$, respectively. We also have $D_{q}x^{n}=[n]_{q}x^{n-1}$ and $D_{q}(1-x)_{q}^{n}=-[n]_{q}(1-qx)_{q}^{n-1}$.

The rest of this paper is organized as follows. In the next section, we give some basic properties of the operators $T_{n,q,\alpha }(f)$, such as the moments and central moments for proving the convergence theorems, the forward difference form of $T_{n,q,\alpha }(f)$ 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 $T_{n,q,\alpha }(f)$.

2 Auxiliary results

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

Lemma 2.1

We have the following equalities:

$$\begin{aligned} T_{n,q,\alpha }(1;x)=1, \quad\quad T_{n,q,\alpha }(t;x)=x. \end{aligned}$$

(7)

Proof

By (5), we have

$$\begin{array}{rcl}{T}_{n,q,\alpha }(1;x)& =& (1-\alpha )\sum _{i=0}^{n-1}{\left[\begin{array}{c}n-1\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-1-i}+\alpha \sum _{i=0}^n{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-i}\\ & =& 1.\end{array}$$

However,

$$\begin{array}{c}{T}_{n,q,\alpha }(t;x)\hfill \\ =(1-\alpha )\sum _{i=0}^{n-1}[(1-\frac{q^{n-1-i}{[i]}_q}{{[n-1]}_q})\frac{{[i]}_q}{{[n]}_q}+\frac{q^{n-1-i}{[i]}_q}{{[n-1]}_q}\frac{{[i+1]}_q}{{[n]}_q}]{\left[\begin{array}{c}n-1\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-1-i}\hfill \\ +\alpha \sum _{i=0}^n\frac{{[i]}_q}{{[n]}_q}{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-i}\hfill \\ =(1-\alpha )\sum _{i=0}^{n-1}\frac{{[i]}_q}{{[n-1]}_q}{\left[\begin{array}{c}n-1\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-1-i}+\alpha \sum _{i=0}^n\frac{{[i]}_q}{{[n]}_q}{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-i}\hfill \\ =(1-\alpha )x+\alpha x=x.\hfill \end{array}$$

Lemma 2.1 is proved. □

Remark 2.2

From Lemma 2.1, we know that the $(\alpha , q)$-Bernstein operators $T_{n,q,\alpha }(f;x)$ reproduce linear functions; that is,

$$\begin{aligned} T_{n,q,\alpha }(at+b;x)=ax+b, \end{aligned}$$

for all real numbers a and b.

We immediately obtain Lemma 2.3 from (5) and Lemma 2.1.

Lemma 2.3

For all functions f and g defined in $[0,1]$, $x\in [0,1]$, real numbers λ, μ defined in $[0,1]$, and $q\in (0,1]$, the following statements hold true.

• (i) Endpoint interpolation: $T_{n,q,\alpha }(f;0)=f(0)$ and $T_{n,q,\alpha }(f;1)=f(1)$.

• (ii) Linearity: $T_{n,q,\alpha }(\lambda f+\mu g;x)= \lambda T_{n,q,\alpha }(f;x)+\mu T_{n,q,\alpha }(g;x)$.

• (iii) Non-negative: For $0\leq \alpha \leq 1$ and $0< q<1$, if f is non-negative on $[0,1]$, so is $(\alpha , q)$-Bernstein operators $T_{n,q,\alpha }(f;x)$.

• (iv) Monotone: For fixed $0\leq \alpha \leq 1$ and $0< q<1$, if $f\geq g$, then $T_{n,q,\alpha }(f;x)\geq T_{n,q,\alpha }(g;x)$.

Lemma 2.4

• (i) The $(\alpha , q)$-Bernstein operators may be expressed in the form

  $$T_{n,q,\alpha }(f;x)=\sum _{r=0}^n((1-\alpha ){\left[\begin{array}{c}n-1\\ r\end{array}\right]}_q{\mathrm{△}}_q^r{g}_0+\alpha {\left[\begin{array}{c}n\\ r\end{array}\right]}_q{\mathrm{△}}_q^r{f}_0)x^r,$$

  (8)

  where ${[\begin{array}{c}n-1\\ n\end{array}]}_q=0$, $\triangle_{q}^{r}f_{j}=\triangle_{q}^{r-1}f_{j+1}-q ^{r-1}\triangle_{q}^{r-1}f_{j}$, $r\geq 1$, with $\triangle_{q}^{0}f _{j}=f_{j}=f ( \frac{[j]_{q}}{[n]_{q}} )$.

• (ii) The higher-order forward difference of $g_{i}$ may be expressed in the form

  $$\begin{aligned} \triangle_{q}^{r}g_{i}= \biggl( 1- \frac{q^{n-i-1}[i]_{q}}{[n-1]_{q}} \biggr) \triangle_{q}^{r}f_{i}+ \frac{q^{n-i-1-r}[i+r]_{q}}{[n-1]_{q}}\triangle _{q}^{r}f_{i+1}, \end{aligned}$$

  (9)

  where $\triangle_{q}^{0}g_{i}=g_{i}$, 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 $r=k\geq 0$. For $r=k+1$, we have

$$\begin{aligned}& \triangle_{q}^{k+1}g_{i} \\& \quad = \triangle_{q}^{k}g_{i+1}-q^{k} \triangle_{q}^{k}g_{i} \\& \quad = \biggl( 1-\frac{q^{n-i-2}[i+1]_{q}}{[n-1]_{q}} \biggr) \triangle_{q} ^{k}f_{i+1}+\frac{q^{n-i-2-k}[i+k+1]_{q}}{[n-1]_{q}}\triangle_{q}^{k}f _{i+2} \\& \quad \quad {} -q^{k} \biggl[ \biggl( 1-\frac{q^{n-i-1}[i]_{q}}{[n-1]_{q}} \biggr) \triangle_{q}^{k}f_{i}+\frac{q^{n-i-k-1}[i+k]_{q}}{[n-1]_{q}} \triangle _{q}^{k}f_{i+1} \biggr] \\& \quad = \biggl[ 1-\frac{q^{n-i-2} ( 1+q[i]_{q} ) }{[n-1]_{q}} \biggr] \triangle_{q}^{k}f_{i+1}- \biggl( 1-\frac{q^{n-i-1}[i]_{q}}{[n-1]_{q}} \biggr) q ^{k}\triangle_{q}^{k}f_{i} \\& \quad \quad {} -\frac{q^{n-i-1}[i+k]_{q}}{[n-1]_{q}}\triangle_{q}^{k}f_{i+1}+ \frac{q ^{n-i-2-k}[i+k]_{q}}{[n-1]_{q}}\triangle_{q}^{k}f_{i+2} \\& \quad = \biggl( 1-\frac{q^{n-i-1}[i]_{q}}{[n-1]_{q}} \biggr) \triangle_{q}^{k+1}f _{i}-\frac{q^{n-i-2}}{[n-1]_{q}}\triangle_{q}^{k}f_{i+1}- \frac{q^{n-i-1}[i+k]_{q}}{[n-1]_{q}}\triangle_{q}^{k}f_{i+1} \\& \quad \quad {} +\frac{q^{n-i-2-k}[i+k+1]_{q}}{[n-1]_{q}}\triangle_{q}^{k}f_{i+2} \\& \quad = \biggl( 1-\frac{q^{n-i-1}[i]_{q}}{[n-1]_{q}} \biggr) \triangle_{q}^{k+1}f _{i}-\frac{q^{n-i-2}[i+k+1]_{q}}{[n-1]_{q}}\triangle_{q}^{k}f_{i+1}+ \frac{q ^{n-i-1-k}[i+k+1]_{q}}{[n-1]_{q}}\triangle_{q}^{k}f_{i+2} \\& \quad = \biggl( 1-\frac{q^{n-i-1}[i]_{q}}{[n-1]_{q}} \biggr) \triangle_{q}^{k+1}f _{i}+\frac{q^{n-i-k-2}[i+k+1]_{q}}{[n-1]_{q}} \bigl( \triangle_{q}^{k}f _{i+2}-q^{k}\triangle_{q}^{k}f_{i+1} \bigr) \\& \quad = \biggl( 1-\frac{q^{n-i-1}[i]_{q}}{[n-1]_{q}} \biggr) \triangle_{q}^{k+1}f _{i}+\frac{q^{n-i-k-2}[i+k+1]_{q}}{[n-1]_{q}}\triangle_{q}^{k+1}f_{i+1}. \end{aligned}$$

This shows that (9) holds when k is replaced by $k+1$, and this completes the proof of Lemma 2.4. □

Since $f [ \frac{[j]_{q}}{[n]_{q}},\frac{[j+1]_{q}}{[n]_{q}},\ldots, \frac{[j+k]_{q}}{[n]_{q}} ] =\frac{[n]_{q}^{k}\triangle_{q}^{k}f _{j}}{q^{\frac{k(2j+k-1)}{2}}[k]_{q}!}=\frac{f^{(k)}(\xi )}{k!}$, where $\xi \in ( \frac{[j]_{q}}{[n]_{q}},\frac{[j+k]_{q}}{[n]_{q}} )$, the q differences of the monomial $x^{k}$ of order greater than k are zero. We see from Lemma 2.4 that, for all $n\geq k$, $T_{n,q,\alpha } ( t^{k};x )$ is a polynomial of degree k. Actually, the $(\alpha , q)$-Bernstein operators are degree-reducing on polynomials; that is, if f is a polynomial of degree m, and then $T_{n,q,\alpha }(f)$ is a polynomial of degree $\leq \min\{m,n\}$. In particular, we have the following results.

Lemma 2.5

Letting $f(t)=t^{k}$, $n-1\geq k\geq 2$, we have

$$\begin{aligned} T_{n,q,\alpha }\bigl(t^{k};x\bigr)=a_{k}x^{k}+a_{k-1}x^{k-1}+ \cdots+a_{1}x+a_{0}, \end{aligned}$$

where $a_{k}=\frac{q^{\frac{k(k-1)}{2}}[n-2]_{q}!}{[n-k]_{q}![n]_{q} ^{k}} \{ (1-\alpha )[n-k]_{q}[n-1+k]_{q}+\alpha [n]_{q}[n-1]_{q} \}$.

Proof

Indeed, from (9) and $\triangle_{q}^{k}f_{j}=\frac{q^{ \frac{k(2j+k-1)}{2}}[k]_{q}!f^{(k)}(\xi )}{k![n]_{q}^{k}}$, we have

$$\begin{aligned} \triangle_{q}^{k}g_{0}=\triangle_{q}^{k}f_{0}+ \frac{q^{n-1-k}[k]_{q}}{[n-1]_{q}}\triangle_{q}^{k}f_{1},\quad\quad \triangle_{q}^{k}f_{0}=\frac{q^{\frac{k(k-1)}{2}}[k]_{q}!}{[n]_{q} ^{k}},\quad\quad \triangle_{q}^{k}f_{1}=\frac{q^{\frac{k(k+1)}{2}}[k]_{q}!}{[n]_{q} ^{k}}. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \triangle_{q}^{k}g_{0}= \biggl( 1+ \frac{q^{n-1}[k]_{q}}{[n-1]_{q}} \biggr) \frac{q ^{\frac{k(k-1)}{2}}[k]_{q}!}{[n]_{q}^{k}}= \frac{[n-1+k]_{q}}{[n-1]_{q}} \frac{q^{\frac{k(k-1)}{2}}[k]_{q}!}{[n]_{q} ^{k}}. \end{aligned}$$

Hence, using (8), we have

$$a_k=[(1-\alpha ){\left[\begin{array}{c}n-1\\ k\end{array}\right]}_q\frac{{[n-1+k]}_q}{{[n-1]}_q}+\alpha {\left[\begin{array}{c}n\\ k\end{array}\right]}_q]\frac{q^{\frac{k(k-1)}{2}}{[k]}_q!}{{[n]}_q^k}.$$

We then obtain the proof of Lemma 2.5 by simple computations. □

Lemma 2.6

The following equalities hold true:

$$\begin{aligned}& T_{n,q,\alpha } \bigl( t^{2};x \bigr) =x^{2}+ \frac{x(1-x)}{[n]_{q}}+\frac{(1- \alpha )q^{n-1}[2]_{q}x(1-x)}{[n]_{q}^{2}}, \end{aligned}$$

(10)

$$\begin{aligned}& T_{n,q,\alpha } \bigl( (t-x)^{2};x \bigr) =\frac{x(1-x)}{[n]_{q}}+ \frac{(1- \alpha )q^{n-1}[2]_{q}x(1-x)}{[n]_{q}^{2}}. \end{aligned}$$

(11)

Proof

For $f(t)=t^{2}$, we have $\triangle_{q}^{0}f_{0}=f_{0}=0$, $\triangle_{q}^{1}f_{0}=f_{1}-f_{0}=\frac{1}{[n]_{q}^{2}}$, $\triangle_{q}^{1}f_{1}=f_{2}-f_{1}=\frac{2q+q^{2}}{[n]_{q}^{2}}$, $\triangle_{q}^{2}f_{0}=\triangle_{q}^{1}f_{1}-q\triangle_{q}^{1}f _{0}=f_{2}-[2]_{q}f_{1}+qf_{0}=\frac{q[2]_{q}}{[n]_{q}^{2}}$, and $\triangle_{q}^{2}f_{1}=f_{3}-[2]_{q}f_{2}+qf_{1}=\frac{q^{3}+q^{4}}{[n]_{q} ^{2}}$. By (9), we have $\triangle_{q}^{0}g_{0}=0$, and

$$\begin{aligned}& \triangle_{q}^{1}g_{0}=\triangle_{q}^{1}f_{0}+ \frac{q^{n-2}}{[n-1]_{q}}\triangle_{q}^{1}f_{1}= \frac{1}{[n]_{q}^{2}}+\frac{2q ^{n-1}+q^{n}}{[n-1]_{q}[n]_{q}^{2}}, \\& \triangle_{q}^{2}g_{0}=\triangle_{q}^{2}f_{0}+ \frac{q^{n-3}[2]_{q}}{[n-1]_{q}}\triangle_{q}^{2}f_{1}= \frac{q[2]_{q}}{[n]_{q} ^{2}}+\frac{[2]_{q} ( q^{n}+q^{n+1} ) }{[n-1]_{q}[n]_{q}^{2}}. \end{aligned}$$

From (8), we have

$$\begin{aligned}& T_{n,q,\alpha } \bigl( t^{2};x \bigr) \\& \quad = (1-\alpha )\triangle_{q}^{0}g_{0}+\alpha \triangle_{q}^{0}f_{0}+ \bigl[ (1-\alpha )[n-1]_{q}\triangle_{q}^{1}g_{0}+ \alpha [n]_{q} \triangle_{q}^{1}f_{0} \bigr] x \\& \quad \quad {} + \biggl[ (1-\alpha )\frac{[n-1]_{q}[n-2]_{q}}{[2]_{q}}\triangle_{q} ^{2}g_{0}+\alpha \frac{[n]_{q}[n-1]_{q}}{[2]_{q}}\triangle_{q}^{2}f _{0} \biggr] x^{2} \\& \quad = \biggl[ \frac{(1-\alpha )[n-1]_{q}}{[n]_{q}^{2}}+\frac{(1-\alpha ) ( 2q^{n-1}+q^{n} ) }{[n]_{q}^{2}}+\frac{\alpha }{[n]_{q}} \biggr] x \\& \quad \quad {} + \biggl[ \frac{(1-\alpha )q[n-1]_{q}[n-2]_{q}}{[n]_{q}^{2}}+\frac{(1- \alpha )[n-2]_{q} ( q^{n}+q^{n+1} ) }{[n]_{q}^{2}}+\frac{ \alpha q[n-1]_{q}}{[n]_{q}} \biggr] x^{2} \\& \quad = \frac{[n]_{q}+(1-\alpha )q^{n-1}[2]_{q}}{[n]_{q}^{2}}x+ \biggl( 1- \frac{1}{[n]_{q}}-\frac{(1-\alpha )q^{n-1}[2]_{q}}{[n]_{q}^{2}} \biggr) x ^{2} \\& \quad = x^{2}+\frac{x(1-x)}{[n]_{q}}+\frac{(1-\alpha )q^{n-1}[2]_{q}x(1-x)}{[n]_{q} ^{2}}. \end{aligned}$$

Hence, (10) is proved. Finally, using Lemma 2.1, we obtain

$$ T_{n,q,\alpha } \bigl( (t-x)^{2};x \bigr) =T_{n,q,\alpha } \bigl( t^{2};x \bigr) -2xT _{n,q,\alpha }(t;x)+x^{2}T_{n,q,\alpha }(1;x)=T_{n,q,\alpha } \bigl( t ^{2};x \bigr) -x^{2}. $$

Then (11) is proved by (10). This completes the proof of Lemma 2.6. □

3 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

Let $\{L_{n}\}$ denote a sequence of monotone linear operators that map a function $f\in C[a,b]$ to a function $L_{n}f\in C[a,b]$, and let $L_{n}f\rightarrow f$ uniformly on $[a,b]$ for $f=1, t$ and $t^{2}$. Then $L_{n}f\rightarrow f$ uniformly on $[a,b]$ for all $f\in C[a,b]$.

Theorem 3.1 leads to the following theorem on the convergence of $(\alpha , q)$-Bernstein operators.

Theorem 3.2

Let $q:=\{q_{n}\}$ denote a sequence such that $q_{n}\in (0,1)$ and $\lim_{n\rightarrow \infty }q_{n}=1$. Then, for any $f\in C[0,1]$ and $\alpha \in [0,1]$, $T_{n,q,\alpha }(f;x)$ converges uniformly to $f(x)$ on $[0,1]$.

Proof

From Lemma 2.1, we see that $T_{n,q,\alpha }(f;x)=f(x)$ for $f(t)=1$ and $f(t)=t$. Since $\lim_{n\rightarrow \infty }q_{n}=1$, we see from (10) that $T_{n,q,\alpha }(f;x)$ converges uniformly to $f(x)$ for $f(t)=t^{2}$ as $n\rightarrow \infty$. It also follows that $T_{n,q,\alpha }$ is a monotone operator by Lemma 2.3; the proof is then completed by applying the Bohman–Korovkin theorem 3.1. □

As we know, the space $C{[0,1]}$ of all continuous functions on $[0,1]$ is a Banach space with sup-norm $\Vert f \Vert :=\sup_{x\in [0,1]} \vert f(x) \vert$. Letting $f\in C{[0,1]}$, the Peetre K functional is defined by $K_{2}(f;\delta ):=\inf_{g\in C^{2}{[0,1]}} \{ \Vert f-g \Vert +\delta \Vert g'' \Vert \}$, where $\delta >0$ and $C^{2}{[0,1]}:=\{g \in C{[0,1]}: g', g''\in C{[0,1]}\}$. By [24], there exists an absolute constant $C>0$, such that

$$ K_{2}(f;\delta )\leq C\omega_{2} ( f;\sqrt{\delta } ) , $$

(12)

where $\omega_{2}(f;\delta ):=\sup_{0< h\leq \delta } \sup_{x,x+h,x+2h\in [0,1]} \vert f(x+2h)-2f(x+h)+f(x) \vert$ is the second-order modulus of smoothness of $f\in C{[0,1]}$.

Theorem 3.3

For $f\in C{[0,1]}$, $\alpha \in [0,1]$, $q\in (0,1)$, we have

$$\begin{aligned} \bigl\vert T_{n,q,\alpha }(f;x)-f(x) \bigr\vert \leq C\omega_{2} \biggl( f;\frac{\sqrt{2[n]_{q}+(1- \alpha )2[2]_{q}q^{n-1}}}{4[n]_{q}} \biggr) , \end{aligned}$$

where C is a positive constant.

Proof

Letting $g\in C^{2}{[0,1]}$, $x,t\in [0,1]$, by Taylor’s expansion we have

$$\begin{aligned} g(t)=g(x)+g'(x) (t-x)+ \int_{x}^{t}(t-u)g''(u) \,du. \end{aligned}$$

Using Lemma 2.1, we obtain

$$\begin{aligned} T_{n,q,\alpha }(g;x)=g(x)+T_{n,q,\alpha } \biggl( \int_{x}^{t}(t-u)g''(u) \,du;x \biggr) . \end{aligned}$$

Thus, we have

$$\begin{aligned} \bigl\vert T_{n,q,\alpha }(g;x)-g(x) \bigr\vert =& \biggl\vert T_{n,q,\alpha } \biggl( \int _{x}^{t}(t-u)g''(u) \,du;x \biggr) \biggr\vert \\ \leq &T_{n,q,\alpha } \biggl( \biggl\vert \int_{x}^{t}(t-u) \bigl\vert g''(u) \bigr\vert \,du \biggr\vert ;x \biggr) \\ \leq &T_{n,q,\alpha } \bigl( (t-x)^{2};x \bigr) \bigl\Vert g'' \bigr\Vert \\ \leq &\frac{[n]_{q}+(1-\alpha )q^{n-1}[2]_{q}}{4[n]_{q}^{2}} \bigl\Vert g'' \bigr\Vert . \end{aligned}$$

(13)

However, using Lemma 2.1, we have

$$\begin{aligned} \bigl\vert T_{n,q,\alpha }(f;x) \bigr\vert \leq \Vert f \Vert . \end{aligned}$$

(14)

Now, (13) and (14) imply

$$\begin{aligned} \bigl\vert T_{n,q,\alpha }(f;x)-f(x) \bigr\vert \leq & \bigl\vert T_{n,q,\alpha }(f-g;x)-(f-g) (x) \bigr\vert + \bigl\vert T_{n,q,\alpha }(g;x)-g(x) \bigr\vert \\ \leq &2 \Vert f-g \Vert + \frac{[n]_{q}+(1-\alpha )q^{n-1}[2]_{q}}{4[n]_{q}^{2}} \bigl\Vert g'' \bigr\Vert . \end{aligned}$$

Hence, taking the infimum on the right-hand side over all $g\in C^{2} {[0,1]}$, we obtain

$$\begin{aligned} \bigl\vert T_{n,q,\alpha }(f;x)-f(x) \bigr\vert \leq 2K_{2} \biggl( f;\frac{[n]_{q}+(1- \alpha )q^{n-1}[2]_{q}}{8[n]_{q}^{2}} \biggr) . \end{aligned}$$

By (12), we obtain

$$\begin{aligned} \bigl\vert T_{n,q,\alpha }(f;x)-f(x) \bigr\vert \leq C\omega_{2} \biggl( f;\frac{\sqrt{2[n]_{q}+(1- \alpha )2[2]_{q}q^{n-1}}}{4[n]_{q}} \biggr) , \end{aligned}$$

where C is a positive constant. Theorem 3.3 is proved. □

Remark 3.4

Letting $q:=\{q_{n}\}$ denote a sequence such that $q_{n}\in (0,1)$ and $\lim_{n\rightarrow \infty }q_{n}=1$, we know that, under the conditions of theorem 3.3, the convergence rate of the operators $T_{n,q, \alpha }(f)$ to f is $1/\sqrt{[n]_{q}}$ as $n\rightarrow \infty$. This convergence rate can be improved depending on the choice of q, at least as fast as $1/\sqrt{n}$.

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 $T_{10,0.9,\alpha }(f;x)$ with $\alpha =0.6$ and $\alpha =0.9$.

Figure 1

Convergence of $T_{n,q,\alpha }(f;x)$ to $f(x)$ for fixed $\alpha =0.9$

Figure 2

Convergence of $T_{n,q,\alpha }(f;x)$ to $f(x)$ for fixed $q=0.9$

4 Shape-preserving properties

The $(\alpha , q)$-Bernstein operators $T_{n,q,\alpha }(f;x)$ have a monotonicity-preserving property.

Theorem 4.1

Let $f\in C{[0,1]}$. If f is a monotonically increasing or monotonically decreasing function on $[0,1]$, so are all its $(\alpha , q)$-Bernstein operators for fixed $q\in (0,1)$ and $\alpha \in [0,1]$.

Proof

From (5), we have

$$T_{n+1,q,\alpha }(f;x)=(1-\alpha )\sum _{i=0}^n{g}_i{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-i}+\alpha \sum _{i=0}^{n+1}{f}_i{\left[\begin{array}{c}n+1\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n+1-i},$$

where $f_{i}=\frac{[i]_{q}}{[n+1]_{q}}$, $g_{i}= ( 1- \frac{q^{n-i}[i]_{q}}{[n]_{q}} ) f_{i}+ \frac{q^{n-i}[i]_{q}}{[n]_{q}}f_{i+1}$. Then the q derivative of $T_{n+1,q,\alpha }(f;x)$ is

$$\begin{array}{c}{D}_q[T_{n+1,q,\alpha }(f;x)]\hfill \\ =(1-\alpha )\sum _{i=0}^n{g}_i{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{D}_q\left[x^i{(1-x)}_q^{n-i}\right]+\alpha \sum _{i=0}^{n+1}{f}_i{\left[\begin{array}{c}n+1\\ i\end{array}\right]}_q{D}_q\left[x^i{(1-x)}_q^{n+1-i}\right],\hfill \end{array}$$

and we denote the first and second parts of the right-hand side of the last equation by $\Lambda_{1}$ and $\Lambda_{2}$, respectively. We then have

$$\begin{array}{c}{\mathrm{\Lambda }}_1\hfill \\ =(1-\alpha )\sum _{i=0}^n{g}_i{\left[\begin{array}{c}n\\ i\end{array}\right]}_q[{[i]}_q{x}^{i-1}{(1-qx)}_q^{n-i}-{[n-i]}_q{x}^i{(1-qx)}_q^{n-i-1}]\hfill \\ =(1-\alpha ){[n]}_q[\sum _{i=1}^n{g}_i{\left[\begin{array}{c}n-1\\ i-1\end{array}\right]}_q{x}^{i-1}{(1-qx)}_q^{n-i}-\sum _{i=0}^{n-1}{g}_i{\left[\begin{array}{c}n-1\\ i\end{array}\right]}_q{x}^i{(1-qx)}_q^{n-i-1}]\hfill \\ =(1-\alpha ){[n]}_q\sum _{i=0}^{n-1}{\left[\begin{array}{c}n-1\\ i\end{array}\right]}_q{x}^i{(1-qx)}_q^{n-i-1}{\mathrm{△}}_q^1{g}_i.\hfill \end{array}$$

Using (9), we obtain

$$\begin{aligned} \triangle_{q}^{1}g_{i}= \biggl( 1- \frac{q^{n-i}[i]_{q}}{[n]_{q}} \biggr) \triangle_{q}^{1}f_{i}+ \frac{q^{n-i-1}[i+1]}{[n]_{q}}\triangle_{q} ^{1}f_{i+1}. \end{aligned}$$

Thus, we have

$$\begin{array}{rcl}{\mathrm{\Lambda }}_1& =& (1-\alpha )\sum _{i=0}^{n-1}[({[n]}_q-q^{n-i}{[i]}_q){\mathrm{△}}_q^1{f}_i+q^{n-i-1}{[i+1]}_q{\mathrm{△}}_q^1{f}_{i+1}]{\left[\begin{array}{c}n-1\\ i\end{array}\right]}_q\\ & & \times x^i{(1-qx)}_q^{n-i-1}.\end{array}$$

(15)

Similarly, we can obtain

$${\mathrm{\Lambda }}_2=\alpha {[n+1]}_q\sum _{i=0}^n{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{x}^i{(1-qx)}_q^{n-i}{\mathrm{△}}_q^1{f}_i.$$

(16)

Therefore, by using (15) and (16), the derivative of $( \alpha , q)$-Bernstein operators $T_{n,q,\alpha }(f;x)$ may be expressed in the form

$$\begin{array}{c}{D}_q[T_{n,q,\alpha }(f;x)]\hfill \\ =(1-\alpha )\sum _{i=0}^{n-1}[({[n]}_q-q^{n-i}{[i]}_q){\mathrm{△}}_q^1{f}_i+q^{n-i-1}{[i+1]}_q{\mathrm{△}}_q^1{f}_{i+1}]{\left[\begin{array}{c}n-1\\ i\end{array}\right]}_q\hfill \\ \times x^i{(1-qx)}_q^{n-i-1}+\alpha {[n+1]}_q\sum _{i=0}^n{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{x}^i{(1-qx)}_q^{n-i}{\mathrm{△}}_q^1{f}_i.\hfill \end{array}$$

Since if f is monotonically increasing on $[0,1]$, the forward differences $\triangle_{q}^{1}f_{i}$ and $\triangle_{q}^{1}f_{i+1}$ are non-negative, and so is $D_{q} [ T_{n,q,\alpha }(f;x) ]$. Hence, $(\alpha , q)$-Bernstein operators $T_{n,q,\alpha }(f;x)$ are monotonically increasing on $[0,1]$ for fixed $q\in (0,1)$ and $\alpha \in [0,1]$. On the contrary, if f is monotonically decreasing on $[0,1]$, then operators $T_{n,q,\alpha }(f;x)$ are monotonically decreasing on $[0,1]$ for fixed $q\in (0,1)$ and $\alpha \in [0,1]$. Theorem 4.1 is proved. □

The $(\alpha , q)$-Bernstein operators $T_{n,q,\alpha }(f;x)$ have a convexity-preserving property

Theorem 4.2

Let $f\in C{[0,1]}$. If f is convex on $[0,1]$, so are all of its $( \alpha , q)$-Bernstein operators $T_{n,q,\alpha }(f;x)$ for fixed $q\in (0,1)$ and $\alpha \in [0,1]$.

Proof

From (5), we obtain

$$\begin{array}{c}{T}_{n+2,q,\alpha }(f;x)\hfill \\ =(1-\alpha )\sum _{i=0}^{n+1}{g}_i{\left[\begin{array}{c}n+1\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n-i+1}+\alpha \sum _{i=0}^{n+2}{f}_i{\left[\begin{array}{c}n+2\\ i\end{array}\right]}_q{x}^i{(1-x)}_q^{n+2-i},\hfill \end{array}$$

where $f_{i}=\frac{[i]_{q}}{[n+2]_{q}}$, $g_{i}= ( 1- \frac{q^{n-i+1}[i]_{q}}{[n+1]_{q}} ) f_{i}+ \frac{q^{n-i+1}[i]_{q}}{[n+1]_{q}}f_{i+1}$. The q-derivative of $T_{n+2,q,\alpha }(f;x)$ can easily obtained by the proof theorem 4.1, which may be expressed as

$$\begin{array}{rcl}{D}_q[T_{n+2,q,\alpha }(f;x)]& =& (1-\alpha ){[n+1]}_q\sum _{i=0}^n{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{x}^i{(1-qx)}_q^{n-i}(g_{i+1}-g_i)\\ & & +\alpha {[n+2]}_q\sum _{i=0}^{n+1}{\left[\begin{array}{c}n+1\\ i\end{array}\right]}_q{x}^i{(1-qx)}_q^{n-i+1}(f_{i+1}-f_i).\end{array}$$

Then we have

$$\begin{array}{c}{D}_q^2[T_{n+2,q,\alpha }(f;x)]\hfill \\ =(1-\alpha ){[n+1]}_q\sum _{i=0}^n{\left[\begin{array}{c}n\\ i\end{array}\right]}_q(g_{i+1}-g_i)D_q\left[x^i{(1-qx)}_q^{n-i}\right]\hfill \\ +\alpha {[n+2]}_q\sum _{i=0}^{n+1}{\left[\begin{array}{c}n+1\\ i\end{array}\right]}_q(f_{i+1}-f_i)D_q\left[x^i{(1-qx)}_q^{n-i-1}\right].\hfill \end{array}$$

By some easy computations, we obtain

$$\begin{array}{rcl}{D}_q^2[T_{n+2,q,\alpha }(f;x)]& =& (1-\alpha ){[n+1]}_q{[n]}_q\sum _{i=0}^{n-1}{\left[\begin{array}{c}n-1\\ i\end{array}\right]}_q{x}^i{(1-q^{2}x)}_q^{n-i-1}{\mathrm{△}}_q^2{g}_i\\ & & +\alpha {[n+2]}_q{[n+1]}_q\sum _{i=0}^n{\left[\begin{array}{c}n\\ i\end{array}\right]}_q{x}^i{(1-q^{2}x)}_q^{n-i}{\mathrm{△}}_q^2{f}_i,\end{array}$$

where $\triangle_{q}^{2}g_{i}= ( 1- \frac{q^{n-i+1}[i]_{q}}{[n+1]_{q}} ) \triangle_{q}^{2}f_{i}+\frac{q ^{n-i-1}[i+2]_{q}}{[n+1]_{q}}\triangle_{q}^{2}f_{i+1}$. By the connection between the second-order q differences and convexity, we know that $\triangle_{q}^{2}f_{i}$ and $\triangle_{q}^{2}f_{i+1}$ are all non-negative since f is convex on $[0,1]$. Hence, we obtain $D_{q}^{2} [ T_{n+2,q,\alpha }(f;x) ] \geq 0$, and then the convexity-preserving property of $T_{n,q,\alpha }(f;x)$. Theorem 4.2 is proved. □

Next, if $f(x)$ is convex, the $(\alpha , q)$-Bernstein operators $T_{n,q,\alpha }(f;x)$, for n fixed, are monotonic in q.

Theorem 4.3

For $0< q_{1}\leq q_{2}\leq 1$, $\alpha \in [0,1]$ and for $f(x)$ convex on $[0,1]$, then $T_{n,q_{2},\alpha }(f;x)\leq T_{n,q_{1},\alpha }(f;x)$.

Proof

In the following main proof of our results, we must introduce a linear polynomial function:

$$\begin{aligned} g(x)=\frac{f_{i+1}-f_{i}}{\frac{[i+1]_{q}}{[n]_{q}}- \frac{[i]_{q}}{[n]_{q}}} \biggl( x-\frac{[i]_{q}}{[n]_{q}} \biggr) +f_{i}, \end{aligned}$$

(17)

where $\frac{[i]_{q}}{[n]_{q}}\leq x<\frac{[i+1]_{q}}{[n]_{q}}$, $f _{i}=f ( \frac{[i]_{q}}{[n]_{q}} )$, $i=0,\ldots,n-1$. Then it is straightforward to check that $g_{i}=g ( \frac{[i]_{q}}{[n-1]_{q}} )$. 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

$$\begin{aligned} T_{n,q,\alpha }(f;x)=(1-\alpha )B_{n-1}^{q}(g;x)+\alpha B_{n}^{q}(f;x). \end{aligned}$$

(18)

We have $B_{n-1}^{q_{2}}(g;x)\leq B_{n-1}^{q_{1}}(g;x)$ and $B_{n}^{q_{2}}(f;x)\leq B_{n}^{q_{1}}(f;x)$, and the desired result is obvious. Theorem 4.3 is proved. □

Finally, if $f(x)$ is convex, we give the monotonicity of $(\alpha , q)$-Bernstein operators $T_{n,q,\alpha }(f; x)$ with n.

Theorem 4.4

If $f(x)$ is convex on $[0,1]$, for fixed $q\in (0,1)$ and $\alpha \in [0,1]$, we have

$$\begin{aligned} T_{n-1,q,\alpha }(f;x)-T_{n,q,\alpha }(f;x)\geq 0\quad (n\geq 2). \end{aligned}$$

Proof

Combining (17) and (18), and the fact that if f and g are convex on $[0,1]$, then

$$\begin{aligned} B_{n-2}^{q}(g;x)\geq B_{n-1}^{q}(g;x), \quad\quad B_{n-1}^{q}(f;x)\geq B_{n}^{q}(f;x) \end{aligned}$$

(see [25]). The desired result is obvious. □

Example 4.5

Letting the convex function $f(x) = 1 - \sin(\pi x)$, $x\in [0,1]$, the graphs of $f(x)$ and $T_{n,0.9,0.9}(f;x)$ with different values of $n=10, 15, 20, 30$ are shown in Fig. 3. Figure 4 shows the graphs of $f(x) = 1 - \sin(\pi x)$ and $T_{10,q,0.9}(f;x)$ with $q=0.6, 0.7, 0.8, 0.9$.

Figure 3

Monotonicity of $T_{n,q,\alpha }(f;x)$ in the parameter n

Figure 4

Monotonicity of $T_{n,q,\alpha }(f;x)$ in the parameter q

5 Conclusion

In this paper, we proposed a new family of generalized Bernstein operators, named $(\alpha , q)$-Bernstein operators, and denoted by $T_{n,q,\alpha }(f)$. 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 $T_{n,q,\alpha }(f)$.