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 $T_{n,q,\alpha }(f)$.

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

$$T_{n,\alpha }=\sum _{i=0}^n{f}_i{p}_{n,i}^{(\alpha )}(x),$$ 1

where $f_i=f(\frac{i}{n})$. For $i=0,1,\dots ,n$, 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 $n\ge 2$.

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 $q\in (0,1]$, $f_i=f(\frac{{[i]}_q}{{[n]}_q})$, $i=0,1,2,\dots ,n$, $p_{1,q,0}^{(\alpha )}(x)=1-x$, $p_{1,q,1}^{(\alpha )}(x)=x$, 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

$$g_i=(1-\frac{q^{n-1-i}{[i]}_q}{{[n-1]}_q})f_i+\frac{q^{n-1-i}{[i]}_q}{{[n-1]}_q}{f}_{i+1}.$$ 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 $0<q\le 1$ and each non-negative integer k, we denote q-integers by ${[k]}_q$, where

$${[k]}_q:=\{\begin{array}{cc}\frac{1-q^k}{1-q},\hfill & q\ne 1,\hfill \\ k,\hfill & q=1.\hfill \end{array}$$

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 $T_{n,q,\alpha }(f)$.

Auxiliary results

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

Lemma 2.1

We have the following equalities:

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

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

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 $0\le \alpha \le 1$ 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 $0\le \alpha \le 1$ and $0<q<1$, if $f\ge g$, then $T_{n,q,\alpha }(f;x)\ge T_{n,q,\alpha }(g;x)$.

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 ${[\begin{array}{c}n-1\\ n\end{array}]}_q=0$, ${\mathrm{△}}_q^r{f}_j={\mathrm{△}}_q^{r-1}{f}_{j+1}-q^{r-1}{\mathrm{△}}_q^{r-1}{f}_j$, $r\ge 1$, with ${\mathrm{△}}_q^0{f}_j=f_j=f(\frac{{[j]}_q}{{[n]}_q})$.
• (ii)
  The higher-order forward difference of $g_i$ may be expressed in the form

  $${\mathrm{△}}_q^r{g}_i=(1-\frac{q^{n-i-1}{[i]}_q}{{[n-1]}_q}){\mathrm{△}}_q^r{f}_i+\frac{q^{n-i-1-r}{[i+r]}_q}{{[n-1]}_q}{\mathrm{△}}_q^r{f}_{i+1},$$ 9

  where ${\mathrm{△}}_q^0{g}_i=g_i$, 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 $r=k\ge 0$. For $r=k+1$, we have

$$\begin{array}{c}{\mathrm{△}}_q^{k+1}{g}_i\hfill \\ ={\mathrm{△}}_q^k{g}_{i+1}-q^k{\mathrm{△}}_q^k{g}_i\hfill \\ =(1-\frac{q^{n-i-2}{[i+1]}_q}{{[n-1]}_q}){\mathrm{△}}_q^k{f}_{i+1}+\frac{q^{n-i-2-k}{[i+k+1]}_q}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_{i+2}\hfill \\ -q^k[(1-\frac{q^{n-i-1}{[i]}_q}{{[n-1]}_q}){\mathrm{△}}_q^k{f}_i+\frac{q^{n-i-k-1}{[i+k]}_q}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_{i+1}]\hfill \\ =[1-\frac{q^{n-i-2}(1+q{[i]}_q)}{{[n-1]}_q}]{\mathrm{△}}_q^k{f}_{i+1}-(1-\frac{q^{n-i-1}{[i]}_q}{{[n-1]}_q})q^k{\mathrm{△}}_q^k{f}_i\hfill \\ -\frac{q^{n-i-1}{[i+k]}_q}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_{i+1}+\frac{q^{n-i-2-k}{[i+k]}_q}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_{i+2}\hfill \\ =(1-\frac{q^{n-i-1}{[i]}_q}{{[n-1]}_q}){\mathrm{△}}_q^{k+1}{f}_i-\frac{q^{n-i-2}}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_{i+1}-\frac{q^{n-i-1}{[i+k]}_q}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_{i+1}\hfill \\ +\frac{q^{n-i-2-k}{[i+k+1]}_q}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_{i+2}\hfill \\ =(1-\frac{q^{n-i-1}{[i]}_q}{{[n-1]}_q}){\mathrm{△}}_q^{k+1}{f}_i-\frac{q^{n-i-2}{[i+k+1]}_q}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_{i+1}+\frac{q^{n-i-1-k}{[i+k+1]}_q}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_{i+2}\hfill \\ =(1-\frac{q^{n-i-1}{[i]}_q}{{[n-1]}_q}){\mathrm{△}}_q^{k+1}{f}_i+\frac{q^{n-i-k-2}{[i+k+1]}_q}{{[n-1]}_q}({\mathrm{△}}_q^k{f}_{i+2}-q^k{\mathrm{△}}_q^k{f}_{i+1})\hfill \\ =(1-\frac{q^{n-i-1}{[i]}_q}{{[n-1]}_q}){\mathrm{△}}_q^{k+1}{f}_i+\frac{q^{n-i-k-2}{[i+k+1]}_q}{{[n-1]}_q}{\mathrm{△}}_q^{k+1}{f}_{i+1}.\hfill \end{array}$$

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

Since $f[\frac{{[j]}_q}{{[n]}_q},\frac{{[j+1]}_q}{{[n]}_q},\dots ,\frac{{[j+k]}_q}{{[n]}_q}]=\frac{{[n]}_q^k{\mathrm{△}}_q^k{f}_j}{q^{\frac{k(2j+k-1)}{2}}{[k]}_q!}=\frac{f^{(k)}(\xi )}{k!}$, 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 $n\ge k$, $T_{n,q,\alpha }(t^k;x)$ 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 $\le min\{m,n\}$. In particular, we have the following results.

Lemma 2.5

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

$$T_{n,q,\alpha }(t^k;x)=a_k{x}^k+a_{k-1}{x}^{k-1}+\cdots +a_{1}x+a_0,$$

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 ${\mathrm{△}}_q^k{f}_j=\frac{q^{\frac{k(2j+k-1)}{2}}{[k]}_q!f^{(k)}(\xi )}{k!{[n]}_q^k}$, we have

$${\mathrm{△}}_q^k{g}_0={\mathrm{△}}_q^k{f}_0+\frac{q^{n-1-k}{[k]}_q}{{[n-1]}_q}{\mathrm{△}}_q^k{f}_1,{\mathrm{△}}_q^k{f}_0=\frac{q^{\frac{k(k-1)}{2}}{[k]}_q!}{{[n]}_q^k},{\mathrm{△}}_q^k{f}_1=\frac{q^{\frac{k(k+1)}{2}}{[k]}_q!}{{[n]}_q^k}.$$

Thus, we obtain

$${\mathrm{△}}_q^k{g}_0=(1+\frac{q^{n-1}{[k]}_q}{{[n-1]}_q})\frac{q^{\frac{k(k-1)}{2}}{[k]}_q!}{{[n]}_q^k}=\frac{{[n-1+k]}_q}{{[n-1]}_q}\frac{q^{\frac{k(k-1)}{2}}{[k]}_q!}{{[n]}_q^k}.$$

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:

$$T_{n,q,\alpha }(t^2;x)=x^2+\frac{x(1-x)}{{[n]}_q}+\frac{(1-\alpha )q^{n-1}{[2]}_{q}x(1-x)}{{[n]}_q^2},$$ 10

$$T_{n,q,\alpha }({(t-x)}^2;x)=\frac{x(1-x)}{{[n]}_q}+\frac{(1-\alpha )q^{n-1}{[2]}_{q}x(1-x)}{{[n]}_q^2}.$$ 11

Proof

For $f(t)=t^2$, we have ${\mathrm{△}}_q^0{f}_0=f_0=0$, ${\mathrm{△}}_q^1{f}_0=f_1-f_0=\frac{1}{{[n]}_q^2}$, ${\mathrm{△}}_q^1{f}_1=f_2-f_1=\frac{2q+q^2}{{[n]}_q^2}$, ${\mathrm{△}}_q^2{f}_0={\mathrm{△}}_q^1{f}_1-q{\mathrm{△}}_q^1{f}_0=f_2-{[2]}_q{f}_1+q{f}_0=\frac{q{[2]}_q}{{[n]}_q^2}$, and ${\mathrm{△}}_q^2{f}_1=f_3-{[2]}_q{f}_2+q{f}_1=\frac{q^3+q^4}{{[n]}_q^2}$. By (9), we have ${\mathrm{△}}_q^0{g}_0=0$, and

$$\begin{array}{c}{\mathrm{△}}_q^1{g}_0={\mathrm{△}}_q^1{f}_0+\frac{q^{n-2}}{{[n-1]}_q}{\mathrm{△}}_q^1{f}_1=\frac{1}{{[n]}_q^2}+\frac{2q^{n-1}+q^n}{{[n-1]}_q{[n]}_q^2},\hfill \\ {\mathrm{△}}_q^2{g}_0={\mathrm{△}}_q^2{f}_0+\frac{q^{n-3}{[2]}_q}{{[n-1]}_q}{\mathrm{△}}_q^2{f}_1=\frac{q{[2]}_q}{{[n]}_q^2}+\frac{{[2]}_q(q^n+q^{n+1})}{{[n-1]}_q{[n]}_q^2}.\hfill \end{array}$$

From (8), we have

$$\begin{array}{c}{T}_{n,q,\alpha }(t^2;x)\hfill \\ =(1-\alpha ){\mathrm{△}}_q^0{g}_0+\alpha {\mathrm{△}}_q^0{f}_0+[(1-\alpha ){[n-1]}_q{\mathrm{△}}_q^1{g}_0+\alpha {[n]}_q{\mathrm{△}}_q^1{f}_0]x\hfill \\ +[(1-\alpha )\frac{{[n-1]}_q{[n-2]}_q}{{[2]}_q}{\mathrm{△}}_q^2{g}_0+\alpha \frac{{[n]}_q{[n-1]}_q}{{[2]}_q}{\mathrm{△}}_q^2{f}_0]x^2\hfill \\ =[\frac{(1-\alpha ){[n-1]}_q}{{[n]}_q^2}+\frac{(1-\alpha )(2q^{n-1}+q^n)}{{[n]}_q^2}+\frac{\alpha }{{[n]}_q}]x\hfill \\ +[\frac{(1-\alpha )q{[n-1]}_q{[n-2]}_q}{{[n]}_q^2}+\frac{(1-\alpha ){[n-2]}_q(q^n+q^{n+1})}{{[n]}_q^2}+\frac{\alpha q{[n-1]}_q}{{[n]}_q}]x^2\hfill \\ =\frac{{[n]}_q+(1-\alpha )q^{n-1}{[2]}_q}{{[n]}_q^2}x+(1-\frac{1}{{[n]}_q}-\frac{(1-\alpha )q^{n-1}{[2]}_q}{{[n]}_q^2})x^2\hfill \\ =x^2+\frac{x(1-x)}{{[n]}_q}+\frac{(1-\alpha )q^{n-1}{[2]}_{q}x(1-x)}{{[n]}_q^2}.\hfill \end{array}$$

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

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

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

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 $L_{n}f\to f$ uniformly on $[a,b]$ for $f=1,t$ and $t^2$. Then $L_{n}f\to f$ 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 ${lim}_{n\to \mathrm{\infty }}{q}_n=1$. 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 ${lim}_{n\to \mathrm{\infty }}{q}_n=1$, we see from (10) that $T_{n,q,\alpha }(f;x)$ converges uniformly to $f(x)$ for $f(t)=t^2$ as $n\to \mathrm{\infty }$. 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 $\parallel f\parallel :={sup}_{x\in [0,1]}|f(x)|$. Letting $f\in C[0,1]$, the Peetre K functional is defined by $K_2(f;\delta ):={inf}_{g\in C^2[0,1]}\{\parallel f-g\parallel +\delta \parallel g^{″}\parallel \}$, where $\delta >0$ and $C^2[0,1]:=\{g\in C[0,1]:g^{\prime },g^{″}\in C[0,1]\}$. By [24], there exists an absolute constant $C>0$, such that

$$K_2(f;\delta )\le C{\omega }_2(f;\sqrt{\delta }),$$ 12

where ${\omega }_2(f;\delta ):={sup}_{0<h\le \delta }{sup}_{x,x+h,x+2h\in [0,1]}|f(x+2h)-2f(x+h)+f(x)|$ 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

$$|T_{n,q,\alpha }(f;x)-f(x)|\le C{\omega }_2(f;\frac{\sqrt{2{[n]}_q+(1-\alpha )2{[2]}_q{q}^{n-1}}}{4{[n]}_q}),$$

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

$$g(t)=g(x)+g^{\prime }(x)(t-x)+{\int }_x^t(t-u)g^{″}(u)du.$$

Using Lemma 2.1, we obtain

$$T_{n,q,\alpha }(g;x)=g(x)+T_{n,q,\alpha }({\int }_x^t(t-u)g^{″}(u)du;x).$$

Thus, we have

$$\begin{array}{rcl}|T_{n,q,\alpha }(g;x)-g(x)|& =& \left|T_{n,q,\alpha }({\int }_x^t(t-u)g^{″}(u)du;x)\right|\\ & \le & T_{n,q,\alpha }\left(\right|{\int }_x^t(t-u)|g^{″}(u)|du|;x)\\ & \le & T_{n,q,\alpha }({(t-x)}^2;x)\parallel g^{″}\parallel \\ & \le & \frac{{[n]}_q+(1-\alpha )q^{n-1}{[2]}_q}{4{[n]}_q^2}\parallel g^{″}\parallel .\end{array}$$ 13

However, using Lemma 2.1, we have

$$|T_{n,q,\alpha }(f;x)|\le \parallel f\parallel .$$ 14

Now, (13) and (14) imply

$$\begin{array}{rcl}|T_{n,q,\alpha }(f;x)-f(x)|& \le & |T_{n,q,\alpha }(f-g;x)-(f-g)(x)|+|T_{n,q,\alpha }(g;x)-g(x)|\\ & \le & 2\parallel f-g\parallel +\frac{{[n]}_q+(1-\alpha )q^{n-1}{[2]}_q}{4{[n]}_q^2}\parallel g^{″}\parallel .\end{array}$$

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

$$|T_{n,q,\alpha }(f;x)-f(x)|\le 2K_2(f;\frac{{[n]}_q+(1-\alpha )q^{n-1}{[2]}_q}{8{[n]}_q^2}).$$

By (12), we obtain

$$|T_{n,q,\alpha }(f;x)-f(x)|\le C{\omega }_2(f;\frac{\sqrt{2{[n]}_q+(1-\alpha )2{[2]}_q{q}^{n-1}}}{4{[n]}_q}),$$

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 ${lim}_{n\to \mathrm{\infty }}{q}_n=1$, 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 $n\to \mathrm{\infty }$. 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 $q=0.9$

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 ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$, 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

$${\mathrm{△}}_q^1{g}_i=(1-\frac{q^{n-i}{[i]}_q}{{[n]}_q}){\mathrm{△}}_q^1{f}_i+\frac{q^{n-i-1}[i+1]}{{[n]}_q}{\mathrm{△}}_q^1{f}_{i+1}.$$

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 ${\mathrm{△}}_q^1{f}_i$ and ${\mathrm{△}}_q^1{f}_{i+1}$ 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 ${\mathrm{△}}_q^2{g}_i=(1-\frac{q^{n-i+1}{[i]}_q}{{[n+1]}_q}){\mathrm{△}}_q^2{f}_i+\frac{q^{n-i-1}{[i+2]}_q}{{[n+1]}_q}{\mathrm{△}}_q^2{f}_{i+1}$. By the connection between the second-order q differences and convexity, we know that ${\mathrm{△}}_q^2{f}_i$ and ${\mathrm{△}}_q^2{f}_{i+1}$ are all non-negative since f is convex on $[0,1]$. Hence, we obtain $D_q^2[T_{n+2,q,\alpha }(f;x)]\ge 0$, 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 $0<q_1\le q_2\le 1$, $\alpha \in [0,1]$ and for $f(x)$ convex on $[0,1]$, then $T_{n,q_2,\alpha }(f;x)\le T_{n,q_1,\alpha }(f;x)$.

Proof

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

$$g(x)=\frac{f_{i+1}-f_i}{\frac{{[i+1]}_q}{{[n]}_q}-\frac{{[i]}_q}{{[n]}_q}}(x-\frac{{[i]}_q}{{[n]}_q})+f_i,$$ 17

where $\frac{{[i]}_q}{{[n]}_q}\le x<\frac{{[i+1]}_q}{{[n]}_q}$, $f_i=f(\frac{{[i]}_q}{{[n]}_q})$, $i=0,\dots ,n-1$. 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

$$T_{n,q,\alpha }(f;x)=(1-\alpha )B_{n-1}^q(g;x)+\alpha B_n^q(f;x).$$ 18

We have $B_{n-1}^{q_2}(g;x)\le B_{n-1}^{q_1}(g;x)$ and $B_n^{q_2}(f;x)\le B_n^{q_1}(f;x)$, 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

$$T_{n-1,q,\alpha }(f;x)-T_{n,q,\alpha }(f;x)\ge 0(n\ge 2).$$

Proof

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

$$B_{n-2}^q(g;x)\ge B_{n-1}^q(g;x),B_{n-1}^q(f;x)\ge B_n^q(f;x)$$

(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

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 $T_{n,q,\alpha }(f)$.

Authors’ contributions

The authors carried out the whole manuscript. All authors read and approved the final manuscript.

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11601266), the Natural Science Foundation of Fujian Province of China (Grant No. 2016J05017) and the Program for New Century Excellent Talents in Fujian Province University.

Competing interests

The authors declare that there have no competing interests.