# 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  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  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  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 . In particular, Ye et al.  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 

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

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

\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  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 . 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 . Temple  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 .

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 

\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

()

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

()

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

()

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

()

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

()

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

()

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

()

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 $$n=2, 3, 4, 5, \text{and } 10$$.

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.

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

()

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.

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.

## Availability of data and materials

All data generated or analyzed during this study are included in this published article.

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## Funding

L.T. Su is supported by the Natural Science Foundation of Fujian Province of China (Grant No. 2020J01783) and the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant No. 2022C001R).

## Author information

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### Contributions

The authors confirm contribution to the paper as follows: study conception and design: L.T. Su, G. Mutlu, B. Çekim ; analysis and interpretation of results: L.T. Su, G. Mutlu, B. Çekim; draft manuscript preparation: G. Mutlu. All authors reviewed the results and approved the final version of the manuscript.

### Corresponding author

Correspondence to Gökhan Mutlu.

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