On the shape-preserving properties of λ-Bernstein operators

Abstract

We investigate the shape-preserving properties of λ-Bernstein operators $B_{n,\lambda } ( f;x )$ that were recently introduced Bernstein-type operators defined by a new Beziér basis with shape parameter $\lambda \in [ -1,1 ]$. For this purpose, we express $B_{n,\lambda } ( f;x )$ as a sum of a classical Bernstein operator and a sum of first order divided differences of f. Using this new representation, we prove that $B_{n,\lambda } ( f;x )$ preserves monotonic functions for all $\lambda \in [ -1,1 ]$. However, we show by a counter example that $B_{n,\lambda } ( f;x )$ does not preserve convex functions for some $\lambda \in [ -1,1 ]$. We present a weaker result for the case $\lambda \in [ 0,1 ]$ for a special class of functions. Finally, we analyze the monotonicity of λ-Bernstein operators with n and show that $B_{n,\lambda } ( f;x )$ is not monotonic with n for some λ if $1/2 <\lambda \leq 1$.

1 Introduction

Bernstein [1] introduced the famous Bernstein operators that are defined by

$$ B_{n} ( f;x ) =\sum_{j=0}^{n}b_{n,j} ( x ) f \biggl( \frac{j}{n} \biggr), $$

(1)

where $f: [ 0,1 ] \rightarrow \mathbb{R}$ is a function, $n\in \mathbb{N}:= \{ 1,2,\ldots \}$, $x\in [ 0,1 ]$ and $b_{n,j} ( x )$ is defined by

$$ b_{n,j} ( x ) := \begin{pmatrix} n \\ j\end{pmatrix} x^{j} ( 1-x ) ^{n-j}, $$

(2)

where $j\in \{ 0,1,2,\ldots,n \}$. Bernstein [1] proved that $B_{n} ( f;x )$ converges to $f(x)$ uniformly on $[ 0,1 ]$ as $n\rightarrow \infty$ for any continuous function $f: [ 0,1 ] \rightarrow \mathbb{R}\mathbbm{.}$

Among all linear positive operators, Bernstein operators are the most studied ones (see the monograph [2] for a survey of studies). This is due to their numerous applications in science and engineering, and also their favorable shape-preserving properties.

Since Bernstein operators possess favorable properties and are widely used in applications, there have been numerous generalizations and variants [3–8]. In particular, Ye et al. [9] introduced a new Bézier basis that is dependent on a shape parameter $\lambda \in [ -1,1 ]$. Using this Bézier basis, new Bernstein-type operators (called λ-Bernstein operators) were introduced [3]

$$ B_{n,\lambda } ( f;x ) :=\sum_{j=0}^{n}b_{n,j} ( \lambda ;x ) f \biggl( \frac{j}{n} \biggr) , $$

where $\lambda \in [ -1,1 ]$ and the Bézier basis is defined by [9]

$$\begin{aligned}& b_{n,0} ( \lambda ;x ) =b_{n,0} ( x ) - \frac {\lambda }{n+1}b_{n+1,1} ( x ) ,\\& b_{n,j} ( \lambda ;x ) =b_{n,j} ( x ) +\lambda \frac {n-2j+1}{n^{2}-1}b_{n+1,j} ( x ) -\lambda \frac {n-2j-1}{n^{2}-1}b_{n+1,j+1} ( x ),\\& b_{n,n} ( \lambda ;x ) =b_{n,n} ( x ) - \frac {\lambda }{n+1}b_{n+1,n} ( x ) , \end{aligned}$$

where $b_{n,j} ( x )$ is given by (2). Note that taking $\lambda =0$, one has the well-known Bernstein operator given by (1). Moreover, introducing the shape parameter λ, one has more modeling flexibility. We refer to [10–14] for more details about λ-Bernstein operators and their variants.

Bernstein operators have favorable shape-preserving properties and studying them is crucial for applications in computer-aided design and computer graphics (see [3, 5, 15] for recent studies). It is well known that Bernstein operators have a convexity-preserving property [2]. Namely, $B_{n} ( f )$ is convex for every n, whenever $f\in C [ 0,1 ]$ is convex. Moreover, Bernstein operators preserve monotonic functions i.e., $B_{n} ( f )$ is a decreasing (increasing) function for all $n\in \mathbb{N}$ whenever $f: [ 0,1 ] \rightarrow \mathbb{R}$ is a decreasing (increasing) function, respectively [2]. Temple [16] investigated the monotonicity of Bernstein operators with n. Namely, if f is a convex function on $[ 0,1 ]$, then $B_{n} ( f;x )$ are monotonic in n, meaning that for all $n\in \mathbb{N}$ and $x\in [ 0,1 ]$ the inequality $B_{n+1} ( f;x ) \leq B_{n} ( f;x )$ holds. The converse of this property also holds [2].

The main purpose of this paper is to investigate the shape-preserving properties of recently introduced λ-Bernstein operators. To this end, we introduce a new representation of $B_{n,\lambda} ( f;x )$ as a sum of a Bernstein operator $B_{n} ( f;x )$ and a sum of first order divided differences of f. With the help of this new expression, we show that λ-Bernstein operators preserve monotonic functions. On the other hand, we show by a counter example that the convexity-preserving property is not satisfied for some $\lambda \in [ -1,1 ]$. However, a weaker result for the convexity-preserving property is proven. Finally, we show that the monotonicity of λ-Bernstein operators with n fails for some $\lambda >1/2$.

2 Preliminaries

Recall that Bernstein basis functions satisfy the following properties [2]

$$\begin{aligned}& b_{n,j} ( x ) =0,\quad \text{if }j>n\text{ or }j< 0, \\& b_{n,j} ( x ) = \biggl( 1-\frac {j}{n+1} \biggr) b_{n+1,j} ( x ) +\frac {j+1}{n+1}b_{n+1,j+1} ( x ) , \end{aligned}$$

(3)

$$\begin{aligned}& \frac {d}{dx}b_{n,j} ( x ) =n \bigl[ b_{n-1,j-1} ( x ) -b_{n-1,j} ( x ) \bigr] . \end{aligned}$$

(4)

Definition 2.1

([2])

Let $x_{1},x_{2},\ldots,x_{r}\in [ 0,1 ]$ be distinct points and f be a real-valued function on $[ 0,1 ]$. Then, the divided difference of f with order $(r-1)$ is defined as

$$ [ x_{1},x_{2},\ldots,x_{r}:f ] :=\sum _{j=1}^{r} \frac {f ( x_{j} ) }{ ( x_{j}-x_{1} ) ( x_{j}-x_{2} ) \cdots ( x_{j}-x_{j-1} ) ( x_{j}-x_{j+1} ) \cdots ( x_{j}-x_{r} ) }. $$

Fix $r\in \mathbb{N}$ and $f: [ 0,1 ] \rightarrow \mathbb{R}$. We say that f is a convex (respectively, concave) function of order r, if all its divided differences with order $(r+1)$ are positive (respectively, negative).

Theorem 2.1

([2])

The identity

$$ \frac {d^{r}}{dx^{r}}B_{n} ( f;x ) = \frac {n!r!}{ ( n-r ) !n^{r}}\sum _{j=0}^{n-r}b_{n-r,j} ( x ) \biggl[ \frac{j}{n},\frac{j+1}{n},\ldots, \frac{j+r}{n} :f \biggr] , $$

(5)

holds for any $f: [ 0,1 ] \rightarrow \mathbb{R}$ and $r\in \{ 0,1,\ldots,n \}$.

Corollary 2.1

([2])

Bernstein operators preserve convexities of all orders. In particular, $B_{n} ( f )$ is decreasing (increasing) for every n whenever f is a decreasing (increasing) function on $[ 0,1 ]$, respectively. Similarly, $B_{n} ( f )$ is convex (concave) for every n whenever f is a convex (concave) function on $[ 0,1 ]$, respectively.

Theorem 2.2

([2])

Bernstein operators satisfy the identity

$$ B_{n+1} ( f;x ) -B_{n} ( f;x ) =- \frac{x ( 1-x ) }{n ( n+1 ) } \sum_{j=0}^{n-1}b_{n-1,j} ( x ) \biggl[ \frac{j}{n}, \frac{j+1}{n+1},\frac{j+1}{n}:f \biggr] $$

for $f: [ 0,1 ] \rightarrow \mathbb{R}$, $n\in \mathbb{N}$, $x \in [ 0,1 ]$.

Corollary 2.2

([2])

If $f: [ 0,1 ] \rightarrow \mathbb{R}$ is convex, then $B_{n+1} ( f;x ) \leq B_{n} ( f;x )$ for all $n\in \mathbb{N}$, $x\in [ 0,1 ]$.

Lemma 2.1

([3])

λ-Bernstein operators satisfy

$$\begin{aligned}& B_{n,\lambda } ( 1;x ) =1,\\& B_{n,\lambda } ( t;x ) =x+\lambda \frac{1-2x+x^{n+1}- ( 1-x ) ^{n+1}}{n ( n-1 ) },\\& B_{n,\lambda } \bigl( t^{2};x \bigr) =x^{2}+ \frac{x ( 1-x ) }{n}+\lambda \biggl[ \frac{2x-4x^{2}+2x^{n+1}}{n ( n-1 ) }+ \frac{x^{n+1}+ ( 1-x ) ^{n+1}-1}{n^{2} ( n-1 ) } \biggr] . \end{aligned}$$

Theorem 2.3

([3])

If $f\in C [ 0,1 ]$ and $\lambda \in [ -1,1 ]$, then $B_{n,\lambda } ( f;x )$ converges to $f(x)$ uniformly on $[ 0,1 ]$ as $n\rightarrow \infty$.

3 Main results

From now on, we will use the notation $f_{j}:=f ( \frac{j}{n} )$ for $j=0,1,2,\ldots,n$.

Lemma 3.1

We can write $B_{n,\lambda } ( f;x )$ in the following form

$$ B_{n,\lambda } ( f;x ) =B_{n} ( f;x ) +\lambda \sum _{j=1}^{n}\frac {n-2j+1}{n^{2}-1}b_{n+1,j} ( x ) [ f_{j} -f_{j-1} ] . $$

(6)

Proof

By definition of λ-Bernstein operators, we have

$$\begin{aligned} B_{n,\lambda } ( f;x ) &=\sum_{j=0}^{n}b_{n,j} ( x ) f_{j} +\lambda \sum_{j=1}^{n} \frac {n-2j+1}{n^{2}-1}b_{n+1,j} ( x ) f_{j} \\ &\quad{} -\lambda \sum_{j=0}^{n-1} \frac {n-2j-1}{n^{2}-1}b_{n+1,j+1} ( x ) f_{j} \\ &=B_{n} ( f;x ) +\lambda \sum_{j=1}^{n} \frac {n-2j+1}{n^{2}-1}b_{n+1,j} ( x ) f_{j} \\ &\quad{} -\lambda \sum_{j=1}^{n} \frac {n-2j+1}{n^{2}-1}b_{n+1,j} ( x ) f_{j-1}. \end{aligned}$$

The last equation easily implies (6) and the proof is completed. □

Lemma 3.2

λ-Bernstein operators satisfy the identity

$$\begin{aligned} \frac {d}{dx}B_{n,\lambda } ( f;x ) &= \sum _{j=0}^{n-1} [ f_{j+1} -f_{j} ] b_{n,j} ( x ) \biggl[ n-j+ \lambda \frac {n-2j-1}{n-1} \biggr] \\ &\quad {}+\sum_{j=0}^{n-1} [ f_{j+1} -f_{j} ] b_{n,j+1} ( x ) \biggl[ j+1-\lambda \frac {n-2j-1}{n-1} \biggr], \end{aligned}$$

(7)

for all $f: [ 0,1 ] \rightarrow \mathbb{R}$ and $\lambda \in [ -1,1 ]$.

Proof

Differentiating the expression (6) and using (4) and (5), we obtain

$$\begin{aligned} \frac {d}{dx}B_{n,\lambda } ( f;x ) &=\sum _{j=0}^{n-1}b_{n-1,j} ( x ) \biggl[ \frac{j}{n},\frac{j+1}{n}:f \biggr] \\ &\quad {}+\lambda \sum_{j=1}^{n} \frac {n-2j+1}{n-1} \bigl( b_{n,j-1} ( x ) -b_{n,j} ( x ) \bigr) ( f_{j}-f_{j-1} ). \end{aligned}$$

Using the property (3), we obtain

$$\begin{aligned} \frac {d}{dx}B_{n,\lambda } ( f;x ) &=n\sum _{j=0}^{n-1} \biggl[ \biggl( 1-\frac {j}{n} \biggr) b_{n,j} ( x ) + \frac {j+1}{n}b_{n,j+1} ( x ) \biggr] ( f_{j+1}-f_{j} ) \\ &\quad {}+\lambda \sum_{j=0}^{n-1} \frac {n-2j-1}{n-1} \bigl( b_{n,j} ( x ) -b_{n,j+1} ( x ) \bigr) ( f_{j+1}-f_{j} ), \end{aligned}$$

which gives (7) and completes the proof. □

Remark 3.1

Taking $\lambda =0$ in equation (7), we arrive at equation (5) for $r=1$.

Theorem 3.1

λ-Bernstein operators preserve monotonic functions for all $\lambda \in [ -1,1 ]$, i.e., $B_{n,\lambda } ( f )$ is decreasing (increasing) for all $n\in \mathbb{N}$ and $\lambda \in [ -1,1 ]$ whenever $f: [ 0,1 ] \rightarrow \mathbb{R}$ is decreasing (increasing), respectively.

Proof

Let $f: [ 0,1 ] \rightarrow \mathbb{R}$ be an increasing function. Then, for all distinct points $u,v\in [ 0,1 ]$, one has

$$ [ u, v:f ] = \frac {f ( v ) -f ( u ) }{v-u}>0. $$

(8)

Since $0\leq j\leq n-1$, it easily follows that $-1\leq 1-\frac{2j}{n-1}\leq 1$. Using $-1\leq \lambda \leq 1$, one easily obtains $-1\leq -\lambda ( 1-\frac{2j}{n-1} ) \leq 1$. As a result, we can write

$$ 0\leq n-j-1\leq n-j+\lambda \biggl( 1-\frac{2j}{n-1} \biggr), $$

(9)

and

$$ 0\leq ( j+1 ) -1\leq ( j+1 ) -\lambda \biggl( 1- \frac{2j}{n-1} \biggr). $$

(10)

Using Lemma 3.2, it follows from (8), (9), and (10) that $\frac {d}{dx}B_{n,\lambda } ( f;x ) >0$ and thus $B_{n,\lambda } ( f;x )$ is increasing. Analogously, one can prove that if f is decreasing, then so is $B_{n,\lambda } ( f;x )$. □

Lemma 3.3

λ-Bernstein operators satisfy

$$\begin{aligned} \frac {d^{2}}{dx^{2}}B_{n,\lambda } ( f;x ) &=\lambda \frac {n ( n+1 ) }{n-1} \bigl\{ b_{n-1,0} ( x ) ( f_{0}-f_{1} ) +b_{n-1,n-1} ( x ) ( f_{n}-f_{n-1} ) \bigr\} \\ &\quad{} +n\sum_{j=0}^{n-2} ( f_{j+2}-2f_{j+1}+f_{j} ) b_{n-1,j} ( x ) \biggl( n-j-1+\lambda \frac {n-2j-3}{n-1} \biggr) \\ &\quad{} +n\sum_{k=0}^{n-2} ( f_{j+2}-2f_{j+1}+f_{j} ) b_{n-1,j+1} ( x ) \biggl( j+1-\lambda \frac {n-2j-1}{n-1} \biggr), \end{aligned}$$

for all $f: [ 0,1 ] \rightarrow \mathbb{R}$ and $\lambda \in [ -1,1 ]$.

Proof

Differentiating (7) and using (4) one has

$$\begin{aligned} \frac {d^{2}}{dx^{2}}B_{n,\lambda } ( f;x ) ={}&n\sum_{j=0}^{n-1} ( f_{j+1}-f_{j} ) \bigl( b_{n-1,j-1} ( x ) -b_{n-1,j} ( x ) \bigr) \biggl( n-j+ \lambda \frac {n-2j-1}{n-1} \biggr) \\ & {} +n\sum_{j=0}^{n-1} ( f_{j+1}-f_{j} ) \bigl( b_{n-1,j} ( x ) -b_{n-1,j+1} ( x ) \bigr) \biggl( j+1- \lambda \frac {n-2j-1}{n-1} \biggr) . \end{aligned}$$

The last equation can be written as

$$\begin{aligned} \frac {d^{2}}{dx^{2}}B_{n,\lambda } ( f;x ) &=n\sum _{j=0}^{n-2} ( f_{j+2}-f_{j+1} ) b_{n-1,j} ( x ) \biggl( n-j-1+\lambda \frac {n-2j-3}{n-1} \biggr) \\ &\quad{} +n\sum_{j=0}^{n-1} ( f_{j}-f_{j+1} ) b_{n-1,j} ( x ) \biggl( n-j-1+ \lambda \frac {n-2j-3}{n-1} \biggr) \\ &\quad{} +n\sum_{j=0}^{n-1} ( f_{j+1}-f_{j} ) b_{n-1,j} ( x ) \biggl( j-\lambda \frac {n-2j+1}{n-1} \biggr) \\ &\quad{} +n\sum_{j=0}^{n-2} ( f_{j}-f_{j+1} ) b_{n-1,j+1} ( x ) \biggl( j+1- \lambda \frac {n-2j-1}{n-1} \biggr). \end{aligned}$$

Similarly, shifting the index in the third sum completes the proof. □

Remark 3.2

Taking $\lambda =0$ in Lemma 3.3, we obtain (5) for $r=2$. Moreover, if $0 \leq j\leq n-2$, then it is obvious that

$$ -1\leq 1-\frac{2 ( j+1 ) }{n-1}\leq 1. $$

Since $-1\leq \lambda \leq 1$, it immediately follows that

$$ -1\leq \lambda \biggl( 1-\frac{2 ( j+1 ) }{n-1} \biggr) \leq 1, $$

and thus

$$ 0\leq n-j-2\leq n-j-1+\lambda \frac {n-2j-3}{n-1}. $$

Similarly, one can show that

$$ 0\leq j+1-\lambda \frac {n-2j-1}{n-1}. $$

Let f $: [ 0,1 ] \rightarrow \mathbb{R}$ be a convex function. Then, all divided differences $[ x_{1},x_{2},x_{3}:f ]$ are positive. It easily follows that

$$ \frac{2}{n^{2}} \biggl[ \frac{j}{n}, \frac{j+1}{n}, \frac{j+2}{n}:f \biggr] =f_{j+2}-2f_{j+1}+f_{j}>0. $$

As a result

$$\begin{aligned} &n\sum_{j=0}^{n-2} ( f_{j+2}-2f_{j+1}+f_{j} ) \biggl[ b_{n-1,j} ( x ) \biggl( n-j-1+\lambda \frac {n-2j-3}{n-1} \biggr) \\ &\quad{} +b_{n-1,j+1} ( x ) \biggl( j+1-\lambda \frac {n-2j-1}{n-1} \biggr) \biggr] >0. \end{aligned}$$

However, the term

$$ \lambda \frac {n ( n+1 ) }{n-1} \biggl\{ b_{n-1,0} ( x ) \biggl[ f ( 0 ) -f \biggl( \frac{1}{n} \biggr) \biggr] +b_{n-1,n-1} ( x ) \biggl[ f ( 1 ) -f \biggl( \frac{n-1}{n} \biggr) \biggr] \biggr\} $$

can be negative or positive (since $-1\leq \lambda \leq 1$), which may cause $\frac {d^{2}}{dx^{2}}B_{n,\lambda } ( f;x ) <0$. We demonstrate this with an example.

Example 3.1

Consider the convex function $f(t)=t^{2}$ on $[ 0,1 ]$. From Lemma 2.1, we obtain

$$ \frac {d^{2}}{dx^{2}}B_{n,\lambda } \bigl( t^{2};x \bigr) =2- \frac{2}{n} +\lambda \frac{-8+ ( n+1 ) ( 2n+1 ) x^{n-1}+ ( n+1 ) ( 1-x ) ^{n-1}}{n ( n-1 ) }. $$

(11)

Taking $\lambda =1$ and $n=2$ in the last equation, we have

$$ \frac {d^{2}}{dx^{2}}B_{2,1} \bigl( t^{2};x \bigr) =6x- \frac{3}{2}, $$

and it is obvious that $B_{2,1} ( t^{2};x )$ is convex on the interval $( \frac {1}{4},1 )$, whereas it is concave on the interval $( 0,\frac {1}{4} )$. Therefore, $B_{n,1} ( f;x )$ does not preserve convexity. Similarly, taking $\lambda =-1$ in equation (11), we give the intervals where $B_{n,-1} ( t^{2};x )$ is convex in Table 1 for different values of n. From the table, it can be observed that $B_{n,-1} ( t^{2};x )$ is not convex on $[ 0,1 ]$ for $n\leq 50$ and thus $B_{n,-1} ( f )$ does not preserve convexity. We can see this in Fig. 1 for $n=2, 3, 4, 5, \text{and } 10$.

Figure 1

Graphs of $B_{n,-1} ( t^{2};x )$ for $n=2, 3, 4, 5, \text{and } 10$

Table 1 The intervals where $B_{n,-1} ( t^{2};x )$ is convex for corresponding values of n

Solving the inequality

$$ \frac {d^{2}}{dx^{2}}B_{n,\lambda } \bigl( t^{2};x \bigr) >0 $$

for different values of λ and n, we see more examples in which $B_{n,\lambda } ( t^{2};x )$ is not convex on $[ 0,1 ]$. We collect these examples in Table 2, which shows the intervals where $B_{n,\lambda } ( t^{2};x )$ is convex for different values of λ and n.

Table 2 The intervals where $B_{n,\lambda } ( t^{2};x )$ is convex for the corresponding values of λ and n

The numerical data shows us that for $\lambda =-\frac{1}{2}$, $B_{n,\lambda } ( t^{2};x )$ is convex on $[ 0,1 ]$ only for $n\geq 5$, for $\lambda =\frac{1}{2}$, $B_{n,\lambda } ( t^{2};x )$ is convex on $[ 0,1 ]$ only for $n\geq 3$. Similarly, we see that $B_{n,\lambda } ( f )$ does not preserve convexity for $\lambda =-\frac{15}{16}$ and $\lambda =-\frac{35}{36}$ when we look at the simple example $f(t)=t^{2}$. Therefore, we conclude the next result.

Remark 3.3

λ-Bernstein operators do not preserve convexity of functions for certain $\lambda \in [ -1,1 ]$.

We have seen that λ-Bernstein operators do not preserve convexity in general. However, we see that in some special cases, they preserve convexity as a result of the representation of $\frac {d^{2}}{dx^{2}}B_{n,\lambda } ( f;x )$ given in Lemma 3.3.

Theorem 3.2

If $f: [ 0,1 ] \rightarrow \mathbb{R}$ is a convex function that is nonincreasing on $( 0,x_{0} )$ and nondecreasing on $( x_{0},1 )$ for an interior point $x_{0}\in ( 0,1 )$, then $B_{n,\lambda } ( f )$ is also convex for all $\lambda \in [ 0,1 ]$ and $n>n_{0}$ where $n_{0}$ is dependent on $x_{0}$.

Proof

From Remark 3.2, it is enough to show that

$$ \lambda \frac {n ( n+1 ) }{n-1} \biggl\{ b_{n-1,0} ( x ) \biggl[ f ( 0 ) -f \biggl( \frac{1}{n} \biggr) \biggr] +b_{n-1,n-1} ( x ) \biggl[ f ( 1 ) -f \biggl( \frac{n-1}{n} \biggr) \biggr] \biggr\} \geq 0. $$

(12)

If $x_{0}<\frac{1}{2}$, then we can choose n such that $\frac{1}{n}< x_{0}$. Then, f is nonincreasing on the interval $( 0,\frac{1}{n} )$ and nondecreasing on the interval $( \frac{n-1}{n},1 )$ for all such $n\in \mathbb{N}$ and thus

$$ f ( 0 ) -f \biggl( \frac{1}{n} \biggr) \geq 0, \qquad f ( 1 ) -f \biggl( \frac{n-1}{n} \biggr) \geq 0. $$

(13)

Therefore, we have the inequality (12) for all n such that $\frac{1}{n}< x_{0}$ and for all $\lambda \in [ 0,1 ]$. Similarly if $x_{0}>\frac{1}{2}$, then we can choose n such that $1-\frac{1}{n}>x_{0}$. Then, f is nonincreasing on $( 0,\frac{1}{n} )$ and nondecreasing on $( \frac{n-1}{n},1 )$ for all such $n\in \mathbb{N}$, which yields the inequality (12) for $\lambda \in [ 0,1 ]$. Finally, if $x_{0}=\frac{1}{2}$, then the inequality (12) holds for all $n\geq 2$ and $\lambda \in [ 0,1 ]$. □

Definition 3.1

([17])

Let $f: [ 0,1 ] \rightarrow \mathbb{R}$ be continuous. f is called quasiconvex on $[ 0,1 ]$ if

$$ f \bigl( \eta u+ ( 1-\eta ) v \bigr) \leq \max \bigl\{ f(u),f(v) \bigr\} ,\quad \forall u,v,\eta \in [ 0,1 ] . $$

f is quasiconvex on $[ 0,1 ]$ iff f is nonincreasing and nondecreasing on the intervals $[ 0,c ]$ and $[ c,1 ]$, respectively, where $c\in [ 0,1 ]$. Obviously, every nondecreasing, nonincreasing or convex function is quasiconvex on $[ 0,1 ]$.

Remark 3.4

Note that in the hypothesis of the last theorem, we have excluded the cases $x_{0}=0$ or $x_{0}=1$. This is because if f is nondecreasing or nonincreasing on $[ 0,1 ]$, we can not have both inequalities in (13). If f is a function that satisfies the hypothesis of the last theorem, then f is a quasiconvex function. Therefore, the assertion of the last theorem does not hold for all quasiconvex functions.

Now, we investigate the monotonicity of $B_{n,\lambda } ( f;x )$ with n. Namely, we try to answer the question “is the inequality $B_{n+1,\lambda } ( f;x ) \leq B_{n,\lambda } ( f;x )$ satisfied for every $n\in \mathbb{N}$ and $x\in [ 0,1 ]$ for fixed $\lambda \in [ -1,1 ]$ if $f: [ 0,1 ] \rightarrow \mathbb{R}$ is an arbitrary convex function?” Again, we consider $f(t)=t^{2}$ and check if this property is satisfied. Basically, from Lemma 2.1 the problem reduces to checking when the inequality

$$\begin{aligned} &\frac{x ( 1-x ) }{n+1}+\lambda \biggl[ \frac{2x-4x^{2}+2x^{n+2}}{n ( n+1 ) }+ \frac{x^{n+2}+ ( 1-x ) ^{n+2}-1}{ ( n+1 ) ^{2}n} \biggr] \\ &\quad \leq \frac{x ( 1-x ) }{n}+\lambda \biggl[ \frac{2x-4x^{2}+2x^{n+1}}{n ( n-1 ) }+ \frac{x^{n+1}+ ( 1-x ) ^{n+1}-1}{n^{2} ( n-1 ) } \biggr] \end{aligned}$$

(14)

is satisfied. We solve this inequality using computer algebra and obtain the data given in Table 3. We have observed that for different negative values of λ, the inequality (14) holds for all $n\in \mathbb{N}$ and $x\in [ 0,1 ]$. As for the positive values of λ, we have seen that the inequality (14) is satisfied for all $n\in \mathbb{N}$ and $x\in [ 0,1 ]$ if $\lambda \leq 1/2$. However, if $\lambda >1/2$, then the inequality (14) does not hold for some x. In Table 3, we give the solutions of the inequality (14) for the corresponding values of n and λ.

Table 3 Solutions of the inequality (14) for the corresponding values of n and λ

From these observations, the next result easily follows.

Remark 3.5

λ-Bernstein operators do not have monotonicity property with n for some $\lambda >1/2$. Namely, the inequality

$$ B_{n+1,\lambda } ( f;x ) \leq B_{n,\lambda } ( f;x ) $$

does not hold for all $n\in \mathbb{N}$ and $x\in [ 0,1 ]$ for some fixed $1/2<\lambda \leq 1$ and arbitrary convex function $f\in C [ 0,1 ]$.

4 Conclusion

In this study, the shape-preserving properties of recently introduced λ-Bernstein operators $B_{n,\lambda } ( f;x )$ have been revealed. These properties are fundamental for the applications in computer graphics and computer-aided design. It has been seen that the monotonicity-preserving property is satisfied for every $\lambda \in [ -1,1 ]$. However, it has been demonstrated with a counter example that the convexity-preserving property fails for some $\lambda \in [ -1,1 ]$. In this case, it has been proven that for a special class of convex functions, $B_{n,\lambda } ( f;x )$ is convex for $\lambda \in [ 0,1 ]$ and $n>n_{0}$ (see Theorem 3.2). Furthermore, it has been shown with a counter example that the monotonicity of λ-Bernstein operators with n also fails for $\lambda > 1/2$. For further studies, a special class of functions for which λ-Bernstein operators preserve convexity for every $\lambda \in [ -1,1 ]$ can be investigated. Moreover, as for the monotonicity property of λ-Bernstein operators with n, we have shown that this property is not satisfied, at least for some $\lambda > 1/2$. However, we were not able to obtain results for the other cases of λ. For this reason, it could be interesting to investigate whether this property is satisfied for $-1 \leq \lambda \leq 1/2$ or not.