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<h1>A Stancu type extension of Cheney and Sharma operators</h1>
<p class="authors">
<span class="author">Tuğba Bostancı\(^\ast \),Gülen Başcanbaz-Tunca\(^\ast \)</span>
</p>
<p class="date">September 22, 2017. Accepted: July 3, 2018. Published online: February 17, 2019.</p>
</div>
<p>\(^\ast \)Department of Mathematics, University of Ankara, Turkey, 06100, e-mail: <span class="tt">tbostanci@ankara.edu.tr</span>, <span class="tt"><p>tunca@science.ankara.edu.tr </p>

<div class="abstract"><p> In this paper, we introduce a Stancu type extension of the well known Cheney and Sharma operators. We consider a recurrence relation for the moments of the operators and give a local approximation result via suitable \(K\)-functional. Moreover, we show that each operator preserves the Lipschitz constant and order of a given Lipschitz continuous function. </p>
<p><b class="bf">MSC.</b> 41A36, 41A25. </p>
<p><b class="bf">Keywords.</b> Stancu operator, Cheney-Sharma operator, Lipschitz continuous function, Abel-Jensen equalities. </p>
</div>
</span></p>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(\beta \) be a nonnegative real number and consider the following Abel-Jensen formulas </p>
<div class="displaymath" id="1-a">
  \begin{eqnarray}  \left( u+v+m\beta \right) ^{m} & =& \sum \limits _{k=0}^{m}\tbinom {m}{k}u\left( u+k\beta \right) ^{k-1}[v+\left( m-k\right) \beta ]^{m-k}, \label{1-a} \\ \left( u+v+m\beta \right) ^{m} & =& \sum \limits _{k=0}^{m}\tbinom {m}{k}\left( u+k\beta \right) ^{k}v\left[ v+\left( m-k\right) \beta \right] ^{m-k-1}, \label{1-b} \\ \  \  \  \left( u\! +\! v\right) \left( u\! +\! v\! +\! m\beta \right) ^{m-1}& =& \sum \limits _{k=0}^{m}\tbinom {m}{k}u\left( u+k\beta \right) ^{k-1}v\left[ v+\left( m-k\right) \beta \right] ^{m-k-1}, \label{Jensen} \end{eqnarray}
</div>
<p>where \(u,v\in \mathbb {R} \) and \(m\geq 1\) (see, <i class="itshape">e.g.</i>, <span class="cite">
	[
	<a href="#Stancu-Stoica" >12</a>
	]
</span>). In <span class="cite">
	[
	<a href="#C-S" >5</a>
	]
</span>, Cheney and Sharma generalized the well-known Bernstein polynomials by taking \(\beta \geq 0,\  u=x\  \)and \(v=1-x,\  x\in \lbrack 0,1],\  \)and \(m=n\in \mathbb {N} \) in (<a href="#1-a">1</a>) and (<a href="#Jensen">3</a>) as in the following forms:</p>
<div class="equation" id="Pn,beta Op">
<p>
  <div class="equation_content">
    \begin{equation}  \mathbf{P}_{n}^{\beta }\left( f;x\right) :=\left( 1+n\beta \right) ^{-n}\sum \limits _{k=0}^{n}\tbinom {n}{k}x\left( x+k\beta \right) ^{k-1}\left[ 1\! -\! x+\left( n-k\right) \beta \right] ^{n-k}f\left( \tfrac {k}{n}\right) \label{Pn,beta Op} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p>and</p>
<div class="equation" id="Gn">
<p>
  <div class="equation_content">
    \begin{equation}  G_{n}^{\beta }\left( f;x\right) :=\sum \limits _{k=0}^{n}P_{n,k}^{\beta }\left( x\right) f\left( \tfrac {k}{n}\right) , \label{Gn} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p>where </p>
<div class="equation" id="Pn,k,beta">
<p>
  <div class="equation_content">
    \begin{equation}  P_{n,k}^{\beta }\left( x\right) :=\tfrac {\tbinom {n}{k}x\left( x+k\beta \right) ^{k-1}\left( 1-x\right) \left[ 1-x+\left( n-k\right) \beta \right] ^{n-k-1}}{\left( 1+n\beta \right) ^{n-1}} \label{Pn,k,beta} \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p>for \(f\in C[0,1],\  \)the space of real valued, continuous functions on \([0,1].\  \)Denoting \(e_{v}\left( t\right) :=t^{v},\  t\in \lbrack 0,1], v=0,1,2,\cdots ,\  \)it is obvious that <span class="cite">
	[
	<a href="#C-S" >5</a>
	]
</span> </p>
<div class="equation" id="Gn(e0)">
<p>
  <div class="equation_content">
    \begin{equation}  G_{n}^{\beta }\left( e_{0};x\right) =1. \label{Gn(e0)} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p>Moreover, from <span class="cite">
	[
	<a href="#Stancu-Cismasiu" >10</a>
	]
</span>, we have</p>
<div class="equation" id="Gn(e0,e1">
<p>
  <div class="equation_content">
    \begin{equation}  G_{n}^{\beta }\left( e_{1};x\right) =x. \label{Gn(e0,e1} \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>Since\(\   \beta \geq 0,\  \)these operators are linear and positive and called as Bernstein type Cheney and Sharma operators. In <span class="cite">
	[
	<a href="#C-S" >5</a>
	]
</span>, using the reduction formula </p>
<div class="displaymath" id="a0000000003">
  \begin{equation*}  S\left( k,n,x,y\right) =xS\left( k-1,n,x,y\right) +n\beta S\left( k,n-1,x+\beta ,y\right) , \end{equation*}
</div>
<p>where \(S\) is given by</p>
<div class="equation" id="S(k,n,x,y)">
<p>
  <div class="equation_content">
    \begin{equation}  S\left( k,n,x,y\right) :=\sum \limits _{v=0}^{n}\tbinom {n}{v}\left( x+v\beta \right) ^{v+k-1}\left( y+\left( n-v\right) \beta \right) ^{n-v}, \label{S(k,n,x,y)} \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p>the authors proved uniform convergence of each sequence of operators \(\mathbf{P}_{n}^{\beta }\left( f\right) \  \\ \)and \(G_{n}^{\beta }\left( f\right) \  \)to\(\  f\) on \([0,1]\  \)by taking \(\beta \  \)as a sequence\(\  \)of nonnegative real numbers satisfying \(\beta =o\big( \frac{1}{n}\big) ,\  n\rightarrow \infty \  \)in (<a href="#Pn,beta Op">4</a>)\(\  \)and\(\  \)(<a href="#Gn">5</a>) (see <span class="cite">
	[
	<a href="#C-S" >5</a>
	, 
	Lemma 1
	]
</span>, also, <span class="cite">
	[
	<a href="#Altomare-Campiti" >1</a>
	, 
	pp.
	
	322-326
	]
</span>). Some works concerning these operators are <span class="cite">
	[
	<a href="#Stancu-Cismasiu" >10</a>
	]
</span>, <span class="cite">
	[
	<a href="#Craciun" >6</a>
	]
</span>, <span class="cite">
	[
	<a href="#Stancu-Stoica" >12</a>
	]
</span>, <span class="cite">
	[
	<a href="#Stancu" >11</a>
	]
</span>, <span class="cite">
	[
	<a href="#Stancu-Occorsio" >13</a>
	]
</span>, <span class="cite">
	[
	<a href="#Teodora- Otrocol" >14</a>
	]
</span> and <span class="cite">
	[
	<a href="#BTEII" >2</a>
	]
</span>. It is obvious that \(\mathbf{P}_{n}^{0}=G_{n}^{0}=B_{n},\  \)where \(B_{n}\  \)is the \(n\)-th Bernstein operator. </p>
<p>In <span class="cite">
	[
	<a href="#Stancu-1981" >8</a>
	]
</span>, Stancu constructed the following Bernstein type linear positive operators </p>
<div class="equation" id="Stancu op">
<p>
  <div class="equation_content">
    \begin{equation}  L_{n,r}\left( f;x\right) :=\sum \limits _{k=0}^{n-r}P_{n-r,k}\left( x\right) \left[ \left( 1-x\right) f\left( \tfrac {k}{n}\right) +xf\left( \tfrac {k+r}{n}\right) \right] , \label{Stancu op} \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p>for \(f\in C[0,1],\  \)where \(P_{n,k}\left( x\right) :=\tbinom {n}{k}x^{k}\left( 1-x\right) ^{n-k},\  n\in \mathbb {N} ,\  r\  \)is a non-negative integer parameter with \(n{\gt}2r\  \)(see, also, <span class="cite">
	[
	<a href="#Stancu-1983" >9</a>
	]
</span>). </p>
<p>In <span class="cite">
	[
	<a href="#Yang Xiong Cao" >15</a>
	]
</span>, Yang, Xiong and Cao extended the operators \(L_{n,r}\  \)given\(\  \)by (<a href="#Stancu op">10</a>) to the multivariate setting on a simplex and called them as multivariate Stancu operators. In the work, using elementary method, the authors proved that the multivariate Stancu operators preserves Lipschitz property of the operand. In <span class="cite">
	[
	<a href="#Bustamenta-Quesada" >4</a>
	]
</span>, Bustamante and Quesada gave an asymptotic property for Stancu operators \(L_{n,r}\  \)related to Voronovskaja-type formula. </p>
<p>In the present paper, we consider Stancu operators \(L_{n,r}\  \)in the basis of the Bernstein type Cheney and Sharma operators \(G_{n}^{\beta }\  \)given by (<a href="#Gn">5</a>). For this purpose, we consider </p>
<div class="equation" id="C-S-Stancu">
<p>
  <div class="equation_content">
    \begin{equation}  L_{n,r}^{\beta }\left( f;x\right) :=\sum \limits _{k=0}^{n-r}P_{n-r,k}^{\beta }\left( x\right) \left[ \left( 1-x\right) f\left( \tfrac {k}{n}\right) +xf\left( \tfrac {k+r}{n}\right) \right] , \label{C-S-Stancu} \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p>for \(f\in C[0,1]\  \)and\(\  r\  \)is a non-negative integer parameter with \(n{\gt}2r,\  n\in \mathbb {N} ,\  \)where \(P_{n-r,k}^{\beta }\  \)is given by (<a href="#Pn,k,beta">6</a>) with \(n-r\) in places of \(n\).\(\  \)We shall call these operators as Stancu type extension of Cheney and Sharma operators. For the calculation of moments, we use the same recurrence relationship which is obtained from another quantity that is slightly different from (<a href="#S(k,n,x,y)">9</a>). Namely, the quantity (<a href="#S(k,n,x,y)">9</a>) is closely related to (<a href="#1-a">1</a>), whereas the quantity that we shall use is related with (<a href="#Jensen">3</a>). We study local approximation with the help of suitable \(K-\)functional, and show the preservation of Lipschitz’ constant and order of a Lipschitz continuous function by \(L_{n,r}^{\beta }.\  \)To get approximation results, as in <span class="cite">
	[
	<a href="#C-S" >5</a>
	]
</span>, we take \(\beta ,\  \)as a sequence of positive real numbers such that \(\beta =o( \frac{1}{n}) \   ( n\rightarrow \infty ) .\  \)It is  obvious that \(L_{n,r}^{0}\  \)reduces to the Stancu operator \(L_{n,r}\  \)given by (<a href="#Stancu op">10</a>) and \(L_{n,0}^{\beta }\  \)reduces to the Cheney and Sharma operator (<a href="#Gn">5</a>). </p>
<h1 id="a0000000004">2 Auxiliary Results</h1>
<p>Using the similar technique of <span class="cite">
	[
	<a href="#C-S" >5</a>
	]
</span>, we consider the following quantity <br />\(T\left( k,n,x,y\right) \  \)to get the subsequent recurrence relationship. </p>
<p><div class="lemma_thmwrapper " id="LemmaReduction">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.1</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let \(x,y\in \lbrack 0,1],\  n\in \mathbb {N} ,\) \(k=0,1,...,n,\) and</p>
<div class="equation" id="Sq">
<p>
  <div class="equation_content">
    \begin{equation}  T\left( k,n,x,y\right) :=\sum \limits _{v=0}^{n}\tbinom {n}{v}\left( x+v\beta \right) ^{v+k-1}\left( y+\left( n-v\right) \beta \right) ^{n-v-1}. \label{Sq} \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p>Then one has</p>
<div class="equation" id="Reccurence">
<p>
  <div class="equation_content">
    \begin{equation}  T\left( k,n,x,y\right) =xT\left( k-1,n,x,y\right) +n\beta T\left( k,n-1,x+\beta ,y\right) . \label{Reccurence} \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p>Namely, \(T\  \)satisfies the same reduction formula that <a href="#S(k,n,x,y)" class="eqref">9</a> holds. </p>
<p><div class="proof_wrapper" id="a0000000005">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Direct calculation gives the result. <div class="proof_wrapper" id="a0000000006">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>

  </div>
</div> </p>
<p><div class="conclusion_thmwrapper " id="Conc.">
  <div class="conclusion_thmheading">
    <span class="conclusion_thmcaption">
    Conclusion
    </span>
    <span class="conclusion_thmlabel">2.1</span>
  </div>
  <div class="conclusion_thmcontent">
  <p>For the quantity <a href="#Sq" class="eqref">12</a>, one has the following results: </p>
<p>\(\left( i\right) \  \)From <a href="#Jensen" class="eqref">3</a>, with \(u=x,\  v=y\) and \(m=n,\) it readily follows that</p>
<div class="displaymath" id="a0000000007">
  \begin{equation*}  xyT\left( 0,n,x,y\right) =\left( x+y\right) \left( x+y+n\beta \right) ^{n-1}. \end{equation*}
</div>
<p>\(\left( ii\right) \  \)From <a href="#1-b" class="eqref">2</a>, with \(u=x,\  v=y\) and \(m=n,\  \)it holds   </p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  yT\left( 1,n,x,y\right) =\left( x+y+n\beta \right) ^{n}. \end{equation*}
</div>
<p>\(\left( iii\right) \  \)Recursive application of the formula (<a href="#Reccurence">13</a>) gives that</p>
<div class="displaymath" id="a0000000009">
  \begin{equation*}  T\left( 2,n,x,y\right) =\sum \limits _{v=0}^{n}\tbinom {n}{v}v!\left( x+v\beta \right) \beta ^{v}T\left( 1,n-v,x+v\beta ,y\right) . \end{equation*}
</div>
<p>Moreover, as in <span class="cite">
	[
	<a href="#C-S" >5</a>
	]
</span>, using the fact </p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  v!=\int \limits _{0}^{\infty }e^{-s}s^{v}ds \end{equation*}
</div>
<p>and the binomial formula, one gets</p>
<div class="equation" id="S2">
<p>
  <div class="equation_content">
    \begin{equation}  yT\left( 2,n,x,y\right) =\int \limits _{0}^{\infty }e^{-s}\left[ x\left( x\! +\! y\! +\! n\beta \! +\! s\beta \right) ^{n}\! +\! ns\beta ^{2}\left( x\! +\! y\! +\! n\beta \! +\! s\beta \right) ^{n-1}\right] ds. \label{S2} \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>

  </div>
</div> </p>
<p>Below, making use of (<a href="#Sq">12</a>) and (<a href="#Reccurence">13</a>), we give \(G_{n}^{\beta }\left( e_{2};x\right) \). </p>
<p><div class="lemma_thmwrapper " id="Lemma moments">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.2</span>
  </div>
  <div class="lemma_thmcontent">
  <p>\(\  \)For every \(x\in \lbrack 0,1],\  n\in \mathbb {N} ,\  \)one has</p>
<div class="displaymath" id="a0000000011">
  \begin{equation*}  G_{n}^{\beta }\left( e_{2};x\right) =\tfrac {n-1}{n}\left\{  x\left( x+2\beta \right) A_{n}+x\left( n-2\right) \beta ^{2}B_{n}\right\}  +\tfrac {x}{n}, \end{equation*}
</div>
<p>where</p>
<div class="equation" id="An">
<p>
  <div class="equation_content">
    \begin{equation}  A_{n}=\tfrac {1}{\left( 1+n\beta \right) }\int \limits _{0}^{\infty }e^{-s}\left( 1+\tfrac {s\beta }{1+n\beta }\right) ^{n-2}ds \label{An} \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p>and </p>
<div class="equation" id="Bn">
<p>
  <div class="equation_content">
    \begin{equation}  B_{n}=\tfrac {1}{\left( 1+n\beta \right) ^{2}}\int \limits _{0}^{\infty }se^{-s}\left( 1+\tfrac {s\beta }{1+n\beta }\right) ^{n-3}ds. \label{Bn} \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000012">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> It is easy to see that</p>
<div class="displaymath" id="a0000000013">
  \begin{equation*}  G_{n}^{\beta }\left( e_{2};x\right) =\tfrac {n-1}{n}\tfrac {x\left( 1-x\right) }{\left( 1+n\beta \right) ^{n-1}}T\left( 2,n-2,x+2\beta ,1-x\right) +\tfrac {x}{n}. \end{equation*}
</div>
<p> Using (<a href="#S2">14</a>), \(G_{n}^{\beta }\left( e_{2};x\right) \  \)can be represented as </p>
<div class="displaymath" id="a0000000014">
  \begin{align*} & G_{n}^{\beta }\left( e_{2};x\right)= \\ & =\tfrac {n-1}{n}\tfrac {x}{\left( 1+n\beta \right) ^{n-1}} \int \limits _{0}^{\infty }e^{-s}\left[ \left( x\! \! +\! \! 2\beta \right) \left( 1\! \! +\! \! n\beta \! \! +\! \! s\beta \right) ^{n-2}\! \! +\! \! \left( n\! \! -\! \! 2\right) s\beta ^{2}\left( 1\! \! +\! \! n\beta \! \! +\! \! s\beta \right) ^{n-3}\right] ds \! +\! \tfrac {x}{n} \\ & =\tfrac {n-1}{n}\left\{  x\left( x\! +\! 2\beta \right) A_{n}\! +\! x\left( n\! -\! 2\right) \beta ^{2}B_{n}\right\} \! +\! \tfrac {x}{n}, \end{align*}
</div>
<p>where \(A_{n}\  \)and \(B_{n}\  \)are given by (<a href="#An">15</a>) and (<a href="#Bn">16</a>), respectively. <div class="proof_wrapper" id="a0000000015">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Now, we need to evaluate the limits \(\lim _{n\rightarrow \infty }A_{n}\  \)and \(\lim _{n\rightarrow \infty }B_{n},\  \)when\(\   \beta =o(\frac{1}{n}) .\) </p>
<p><div class="lemma_thmwrapper " id="An-Bn">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2.3</span>
  </div>
  <div class="lemma_thmcontent">
  <p>Let \(\beta \  \)be a sequence of positive real numbers such that \(\beta =o(\frac{1}{n}) .\  \)Then we have </p>
<div class="displaymath" id="a0000000016">
  \begin{equation*}  \lim _{n\rightarrow \infty }A_{n}=1\   \text{and }\lim _{n\rightarrow \infty }B_{n}\  =1. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000017">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> From (<a href="#An">15</a>), writing \(A_{n+2}\) and making change of variable \(u=\frac{s\beta }{1+n\beta }\) we have </p>
<div class="displaymath" id="a0000000018">
  \begin{equation*}  A_{n+2}=\tfrac {1}{\beta }\int \limits _{0}^{\infty }e^{-\frac{\left( 1+\left( n+2\right) \beta \right) }{\beta }u}\left( 1+u\right) ^{n}du. \end{equation*}
</div>
<p>As in <span class="cite">
	[
	<a href="#Stancu-1983" >9</a>
	]
</span>, using the inequalities</p>
<div class="equation" id="estimate">
<p>
  <div class="equation_content">
    \begin{equation}  e^{nu}\left( 1-nu^{2}\right) \leqslant \left( 1+u\right) ^{n}\leqslant e^{nu} \label{estimate} \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p>it follows that</p>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  \tfrac {1}{1+2\beta }-\tfrac {2n\beta ^{2}}{\left( 1+2\beta \right) ^{3}}\leqslant A_{n+2}\leqslant \tfrac {1}{1+2\beta }. \end{equation*}
</div>
<p>Similarly, from (<a href="#Bn">16</a>), for \(B_{n+3}\) we get</p>
<div class="displaymath" id="a0000000020">
  \begin{equation*}  B_{n+3}=\tfrac {1}{\beta ^{2}}\int \limits _{0}^{\infty }e^{-\frac{\left( 1+\left( n+3\right) \beta \right) }{\beta }u}u\left( 1+u\right) ^{n}du. \end{equation*}
</div>
<p>In view of (<a href="#estimate">17</a>), we obtain</p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  \tfrac {1}{\left( 1+3\beta \right) ^{2}}-\tfrac {6n\beta ^{2}}{\left( 1+3\beta \right) ^{4}}\leqslant B_{n+3}\leqslant \tfrac {1}{\left( 1+3\beta \right) ^{2}}. \end{equation*}
</div>
<p>Hence, using the fact \(\beta =\beta _{n}{\gt}0\   \left( n\in \mathbb {N} \right) \  \)satisfying \(\lim _{n\rightarrow \infty }n\beta _{n}=0,\  \)we conclude that \(\lim _{n\rightarrow \infty }A_{n}=1\  \)and \(\lim _{n\rightarrow \infty }B_{n}=1.\) <div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="a0000000023">3 Approximation properties of the Stancu type extension of Cheney and Sharma operators</h1>
<p>In this section, we study some approximation properties of the Stancu type extension of Cheney and Sharma operators \(L_{n,r}^{\beta }\  \)given by (<a href="#C-S-Stancu">11</a>). The moments of the operators can be expressed in terms of the moments of the Cheney and Sharma operators \(G_{n}^{\beta }.\) Namely, we have </p>
<p><div class="lemma_thmwrapper " id="momentsS_CS">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.1</span>
  </div>
  <div class="lemma_thmcontent">
  <p>For every \(x\in \lbrack 0,1],\  n{\gt}2r,\  r\in \mathbb {N} \cup \left\{  0\right\}  ,\  n\in \mathbb {N} ,\  \)we have</p>
<div class="displaymath" id="a0000000024">
  \begin{align*}  L_{n,r}^{\beta }\left( e_{0};x\right) & =1, \\ L_{n,r}^{\beta }\left( e_{1};x\right) & =x, \\ L_{n,r}^{\beta }\left( e_{2};x\right) & =\tfrac {\left( n-r\right) \left( n-r-1\right) }{n^{2}}\left\{  x\left( x+2\beta \right) A_{n-r}+x\left( n-r-2\right) \beta ^{2}B_{n-r}\right\}  \\ & \quad +\tfrac {x}{n^{2}}\left\{  \left( n-r\right) \left( 1+2xr\right) +r^{2}\right\}  , \end{align*}
</div>
<p>where \(A_{n-r}\  \)and \(B_{n-r}\  \)are given by <a href="#An" class="eqref">15</a> and <a href="#Bn" class="eqref">16</a>, respectively. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000025">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Taking (<a href="#Gn">5</a>), (<a href="#Gn(e0)">7</a>), (<a href="#Gn(e0,e1">8</a>), (<a href="#C-S-Stancu">11</a>) and Lemma <a href="#Lemma moments">2.2</a> into consideration, we get  </p>
<div class="displaymath" id="a0000000026">
  \begin{align*}  L_{n,r}^{\beta }\left( e_{0};x\right) & =G_{n-r}^{\beta }\left( e_{0};x\right) =1, \\ L_{n,r}^{\beta }\left( e_{1};x\right) & =\tfrac {n-r}{n}G_{n-r}^{\beta }\left( e_{1};x\right) +\tfrac {rx}{n}G_{n-r}^{\beta }\left( e_{0};x\right) =\left( 1-\tfrac {r}{n}\right) x+\tfrac {r}{n}x =x, \\ L_{n,r}^{\beta }\left( e_{2};x\right) & =\left( \tfrac {n-r}{n}\right) ^{2}G_{n-r}^{\beta }\left( e_{2};x\right) +2xr\tfrac {n-r}{n^{2}}G_{n-r}^{\beta }\left( e_{1};x\right) +\tfrac {xr^{2}}{n^{2}}G_{n-r}^{\beta }\left( e_{0};x\right) \\ & =\tfrac {\left( n-r\right) \left( n-r-1\right) }{n^{2}}\left\{  x\left( x+2\beta \right) A_{n-r}+x\left( n-r-2\right) \beta ^{2}B_{n-r}\right\} \\ & \quad +\tfrac {x}{n^{2}}\left\{  \left( n-r\right) \left( 1+2xr\right) +r^{2}\right\}  , \end{align*}
</div>
<p>where \(A_{n-r}\  \)and \(B_{n-r}\  \)are given by (<a href="#An">15</a>) and (<a href="#Bn">16</a>). <div class="proof_wrapper" id="a0000000027">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Thus, from Lemma <a href="#momentsS_CS">3.1</a> one easily obtains the following result. </p>
<p><div class="corollary_thmwrapper " id="central mom. S_CS">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">3.1</span>
  </div>
  <div class="corollary_thmcontent">
  <p>For every \(x\in \lbrack 0,1],\  n{\gt}2r,\  r\in \mathbb {N} \cup \left\{  0\right\}  ,\  n\in \mathbb {N} ,\  \)we have</p>
<div class="displaymath" id="a0000000028">
  \begin{align*}  L_{n,r}^{\beta }\left( \left( e_{1}-x\right) ;x\right) & =0, \\ L_{n,r}^{\beta }\left( \left( e_{1}-x\right) ^{2};x\right) & =\left( \tfrac {n-r}{n}\right) ^{2}\tfrac {n-r-1}{n-r}\left\{  x\left( x+2\beta \right) A_{n-r}+x\left( n-r-2\right) \beta ^{2}B_{n-r}\right\}  \\ & \quad +\tfrac {2x^{2}r\left( n-r\right) +xr^{2}}{n^{2}}-x^{2} \\ \   \   & =:\delta _{n}\left( x\right) . \end{align*}
</div>

  </div>
</div> </p>
<p>Let us denote the uniform norm on \(C\left[ 0,1\right] \  \)by \(\left\Vert .\right\Vert .\  \)It is not difficult to show that the operators \(L_{n,r}^{\beta }\  \)are bounded from \(C\left[ 0,1\right] \  \)onto itself: </p>
<p><div class="lemma_thmwrapper " id="bounded op.">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">3.2</span>
  </div>
  <div class="lemma_thmcontent">
  <p>For every \(f\in C\left[ 0,1\right] \) we have \( \Vert L_{n,r}^{\beta }\left( f\right) \Vert \leq \left\Vert f\right\Vert \). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000029">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<div class="displaymath" id="a0000000030">
  \begin{eqnarray*}  \left\vert L_{n,r}^{\beta }\left( f;x\right) \right\vert & =& \left\vert \sum \limits _{k=0}^{n-r}P_{n-r,k}^{\beta }\left( x\right) \left[ \left( 1-x\right) f\left( \tfrac {k}{n}\right) +xf\left( \tfrac {k+r}{n}\right) \right] \right\vert \\ & \leq & \sum \limits _{k=0}^{n-r}P_{n-r,k}^{\beta }\left( x\right) \left\vert \left( 1-x\right) f\left( \tfrac {k}{n}\right) +xf\left( \tfrac {k+r}{n}\right) \right\vert \\ & \leq & \sum \limits _{k=0}^{n-r}P_{n-r,k}^{\beta }\left( x\right) \left\{  \left( 1-x\right) \left\vert f\left( \tfrac {k}{n}\right) \right\vert +x\left\vert f\left( \tfrac {k+r}{n}\right) \right\vert \right\}  \\ & \leq & \left\Vert f\right\Vert \sum \limits _{k=0}^{n-r}P_{n-r,k}^{\beta }\left( x\right) \left\{  1-x+x\right\}  \\ & =& \left\Vert f\right\Vert . \end{eqnarray*}
</div>
<p> <div class="proof_wrapper" id="a0000000031">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Recall that the Peetre \(K\)-functional is defined as</p>
<div class="displaymath" id="a0000000032">
  \begin{equation*}  K_{2}\left( f,\delta \right) =\inf \left\{  \left\Vert f-g\right\Vert +\delta \left\Vert g^{\prime \prime }\right\Vert :g\in W^{2}\right\}  , \end{equation*}
</div>
<p>where \(\delta {\gt}0,\) \(W^{2}:=\left\{  g\in C\left[ 0,1\right] :g^{\prime },g^{\prime \prime }\in C\left[ 0,1\right] \right\}  \). From p.177, Theorem 2.4 of <span class="cite">
	[
	<a href="#DeVore-Lor." >7</a>
	]
</span>, there is a positive constant \(C{\gt}0\) such that</p>
<div class="displaymath" id="K2<w2">
  \begin{align}  K_{2}\left( f,\delta \right) \leq C\omega _{2}\left( f,\sqrt{\delta }\right) , \label{K2{\lt}w2} \end{align}
</div>
<p>where </p>
<div class="displaymath" id="a0000000033">
  \begin{equation*}  \omega _{2}\left( f,\delta \right) =\sup \limits _{0\leq h\leq \delta }\sup \limits _{x,\text{ }x+2h\in \left[ 0,1\right] }\left\vert f\left( x+2h\right) -2f\left( x+h\right) +f\left( x\right) \right\vert \end{equation*}
</div>
<p>is the second order modulus of smoothness of \(f\in C\left[ 0,1\right] .\) </p>
<p><div class="theorem_thmwrapper " id="a0000000034">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in C\left[ 0,1\right] ,\  x\in \lbrack 0,1],\  n{\gt}2r,\  r\in \mathbb {N} \cup \left\{  0\right\}  ,\  n\in \mathbb {N} ,\  \)and \(\beta \  \)be a sequence of positive real numbers such that \(\beta =o\left( \frac{1}{n}\right) \). Then</p>
<div class="displaymath" id="a0000000035">
  \begin{equation*} \begin{array}{l} \left\vert L_{n,r}^{\beta }\left( f;x\right) -f\left( x\right) \right\vert \leq C\omega _{2}\left( f,\sqrt{\delta _{n,r}\left( x\right) }\right) ,\end{array}\end{equation*}
</div>
<p>where \(\delta _{n,r}\left( x\right) \) is the same as in Corollary <span class="rm"><a href="#central mom. S_CS">3.1</a></span> and \(C\) is a positive constant. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000036">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> For any function \(g\in W^{2}\  \)and \(x,t\in \lbrack 0,1],\  \)the Taylor formula gives that</p>
<div class="displaymath" id="a0000000037">
  \begin{equation*}  g\left( t\right) =g\left( x\right) +g^{\prime }\left( x\right) \left( t-x\right) +\int \limits _{x}^{t}\left( t-y\right) g^{\prime \prime }\left( y\right) dy. \end{equation*}
</div>
<p>Applying \(L_{n,r}^{\beta }\  \)on both sides of the above formula, linearity and Lemma <a href="#momentsS_CS">3.1</a> give that </p>
<div class="displaymath" id="a0000000038">
  \begin{align*}  \left\vert L_{n,r}^{\beta }\left( g;x\right) -g\left( x\right) \right\vert & \leq \left\Vert g^{\prime \prime }\right\Vert L_{n,r}^{\beta }\left( \int \limits _{x}^{t}\left\vert t-y\right\vert dy;x\right)\leq & \left\Vert g^{\prime \prime }\right\Vert L_{n,r}^{\beta }\left( \left( t-x\right) ^{2};x\right) \\ & =\left\Vert g^{\prime \prime }\right\Vert \delta _{n,r}\left( x\right) . \end{align*}
</div>
<p>Therefore,</p>
<div class="displaymath" id="a0000000039">
  \begin{align*}  \left\vert L_{n,r}^{\beta }\left( f;x\right) -f\left( x\right) \right\vert & \leq \left\vert L_{n,r}^{\beta }\left( f-g;x\right) -\left( f-g\right) \left( x\right) \right\vert +\left\vert L_{n,r}^{\beta }\left( g;x\right) -g\left( x\right) \right\vert \\ & \leq 2\left\Vert f-g\right\Vert +\left\Vert g^{\prime \prime }\right\Vert \delta _{n,r}\left( x\right) . \end{align*}
</div>
<p>Passing to the infimum over all \(g\in W^{2}\) and taking (<a href="#K2<w2">18</a>) into consideration, we obtain</p>
<div class="displaymath" id="a0000000040">
  \begin{align*}  \left\vert L_{n,r}^{\beta }\text{ }\left( f;x\right) -f\left( x\right) \right\vert & \leq 2K_{2}\left( f,\delta _{n,r}\left( x\right) \right) \leq C\omega _{2}\left( f,\sqrt{\delta _{n,r}\left( x\right) }\right) , \end{align*}
</div>
<p>which completes the proof. <div class="proof_wrapper" id="a0000000041">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Next result provides the property of the preservation of Lipschitz’ constant and order of a Lipschitz continuous function by each \(L_{n,r}^{\beta }.\  \)The same result for the Bernstein polynomials was proved by Brown, Elliott and Paget <span class="cite">
	[
	<a href="#B E P" >3</a>
	]
</span>, also, for the Cheney and Sharma operators \(G_{n}^{\beta }\) was obtained in <span class="cite">
	[
	<a href="#BTEII" >2</a>
	]
</span> and for the multivariate Stancu operators was proved in <span class="cite">
	[
	<a href="#Yang Xiong Cao" >15</a>
	]
</span>. </p>
<p>Recall that the class \(\operatorname {Lip}_{M}\left( \alpha ,[0,1]\right) \  \)and the convexity of \(f\in C[0,1]\) are defined, respectively, as </p>
<div class="displaymath" id="a0000000042">
  \begin{equation*} \begin{array}{l} \operatorname {Lip}_{M}\left( \alpha ,[0,1]\right):= \\ :=\left\{  f\in C[0,1]:\left\vert f\left( x\right) -f\left( y\right) \right\vert \leq M\left\vert x-y\right\vert ^{\alpha }\   \forall x,y\in \lbrack 0,1],\  0{\lt}\alpha \leq 1\right\} \end{array}\end{equation*}
</div>
<p>and </p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  f\   \text{is convex}\Leftrightarrow \text{ }f\left( \sum _{k=1}^{n}\alpha _{k}x_{k}\right) \leq \sum _{k=1}^{n}\alpha _{k}f(x_{k}),\quad \forall x_{1},x_{2},\ldots ,x_{n}\in \lbrack 0,1], \end{equation*}
</div>
<p>where \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\geq 0\) satisfying \(\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1\).  </p>
<p><div class="theorem_thmwrapper " id="a0000000044">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(f\in \operatorname {Lip}_{M}\left( \alpha ,[0,1]\right) .\  \)Then \(L_{n,r}^{\beta }\left( f\right) \in \operatorname {Lip}_{M}\left( \alpha ,[0,1]\right) \) for all \(n\in \mathbb {N} \). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000045">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Assume that \(x,y\in \left[ 0,1\right] \) satisfy \(y\geq x.\) Following similar steps used in <span class="cite">
	[
	<a href="#BTEII" >2</a>
	]
</span>, from (<a href="#C-S-Stancu">11</a>) and (<a href="#Jensen">3</a>) \(L_{n,r}^{\beta }\left( f;y\right) \  \)can be written as </p>
<div class="displaymath" id="a0000000046">
  \begin{align*}  L_{n,r}^{\beta }\left( f;y\right)= & \tfrac {1}{\left( 1+\left( n-r\right) \beta \right) ^{n-r-1}}\times \\ & \times \sum \limits _{j=0}^{n-r}\sum \limits _{k=0}^{j}\tbinom {n-r}{j}\tbinom {j}{k}x\left( x+k\beta \right) ^{k-1}\left( y-x\right) \left[ y-x+\left( j-k\right) \beta \right] ^{j-k-1} \\ & \times \left( 1-y\right) \left[ 1-y+\left( n-r-j\right) \beta \right] ^{n-r-j-1} \\ & \times \left\{  \left( 1-y\right) f\big( \tfrac {j}{n}\big) +yf\big( \tfrac {j+r}{n}\big) \right\}  . \end{align*}
</div>
<p>Changing the order of the summations and letting \(j-k=l\  \)in the result, we obtain</p>
<div class="displaymath" id="2.1">
  \begin{align}  L_{n,r}^{\beta }\left( f;y\right) & =\tfrac {1}{\left( 1+\left( n-r\right) \beta \right) ^{n-r-1}} \label{2.1}\times \\ & \quad \times \sum \limits _{k=0}^{n-r}\sum \limits _{l=0}^{n-r-k}\tfrac {\left( n-r\right) !}{\left( n-r-k-l\right) !k!l!}x\left( x+k\beta \right) ^{k-1}\left( y-x\right) \left[ y-x+l\beta \right] ^{l-1} \notag \\ & \quad \times \left( 1-y\right) \left[ 1-y+\left( n-r-k-l\right) \beta \right] ^{n-r-k-l-1} \notag \\ & \quad \times \left\{  \left( 1-y\right) f\big( \tfrac {k+l}{n}\big) +yf\big( \tfrac {k+l+r}{n}\big) \right\}  . \notag \end{align}
</div>
<p>Now, consider the case \(L_{n,r}^{\beta }\left( f;x\right) .\  \)Taking \(u=y-x,\) \(v=1-y\  \)and \(m=n-r-k\  \)in (<a href="#Jensen">3</a>), we can write \(L_{n,r}^{\beta }\left( f;x\right) \  \)as </p>
<div class="displaymath" id="2.2">
  \begin{align}  L_{n,r}^{\beta }\left( f;x\right) & =\tfrac {1}{\left( 1+\left( n-r\right) \beta \right) ^{n-r-1}} \label{2.2} \times \\ & \quad \times \sum \limits _{k=0}^{n-r}\sum \limits _{l=0}^{n-r-k}\tfrac {\left( n-r\right) !}{k!l!\left( n-r-k-l\right) !}x\left( x+k\beta \right) ^{k-1}\left( y-x\right) \left[ y-x+l\beta \right] ^{l-1} \notag \\ & \quad \times \left( 1-y\right) \left[ 1-y+\left( n-r-k-l\right) \beta \right] ^{n-r-k-l-1} \notag \\ & \quad \times \left\{  \left( 1-x\right) f\big( \tfrac {k}{n}\big) +xf\big( \tfrac {k+r}{n}\big) \right\}  . \notag \end{align}
</div>
<p>Thus, subtracting (<a href="#2.2">20</a>) from (<a href="#2.1">19</a>) we get </p>
<div class="displaymath" id="a0000000047">
  \begin{align*}  L_{n,r}^{\beta }\left( f;y\right) \! -\! L_{n,r}^{\beta }\left( f;x\right) & =\tfrac {1}{\left( 1+\left( n-r\right) \beta \right) ^{n-r-1}} \times \\ & \quad \! \! \times \! \! \sum \limits _{k=0}^{n-r}\sum \limits _{l=0}^{n-r-k}\tfrac {\left( n-r\right) !}{k!l!\left( n-r-k-l\right) !}x\big( x\! \! +\! \! k\beta \big) ^{k-1}\big( y\! \! -\! \! x\big) \left[ y\! \! -\! \! x\! \! +\! \! l\beta \right] ^{l-1} \\ & \quad \! \! \times \! \!  \left( 1-y\right) \left[ 1-y+\left( n-r-k-l\right) \beta \right] ^{n-r-k-l-1} \\ & \quad \! \! \times \! \! \left\{  \left( 1\! \! -\! \! y\right) f\big( \tfrac {k+l}{n}\big)\! \! +\! \! yf\big( \tfrac {k+l+r}{n}\big)\! \! - \! \! \left( 1\! \! -\! \! x\right) f\big( \tfrac {k}{n}\big) \! \! -\! \! xf\big( \tfrac {k+r}{n}\big) \right\}  . \end{align*}
</div>
<p>Adding\(\  \)and dropping the terms \(yf(\frac{k}{n})\) and \( xf(\frac{k+l+r}{n}) ,\) the above formula takes to the following form: </p>
<div class="displaymath" id="a0000000048">
  \begin{align*} & L_{n,r}^{\beta }\left( f;y\right) -L_{n,r}^{\beta }\left( f;x\right)= \\ & =\tfrac {1}{\left( 1+\left( n-r\right) \beta \right) ^{n-r-1}}\times \\ & \quad \times \sum \limits _{k=0}^{n-r}\sum \limits _{l=0}^{n-r-k}\tfrac {\left( n-r\right) !}{k!l!\left( n-r-k-l\right) !}x\left( x+k\beta \right) ^{k-1}\left( y-x\right) \left[ y-x+l\beta \right] ^{l-1} \\ & \quad \times \left( 1-y\right) \left[ 1-y+\left( n-r-k-l\right) \beta \right] ^{n-r-k-l-1} \\ & \quad \times \left\{  \left( 1-y\right) \left[ f\left( \tfrac {k+l}{n}\right) -f\left( \tfrac {k}{n}\right) \right] +x\left[ f\left( \tfrac {k+l+r}{n}\right) -f\left( \tfrac {k+r}{n}\right) \right] \right. \\ & \quad \left. +(y-x)\left[ f\left( \tfrac {k+l+r}{n}\right) -f\left( \tfrac {k}{n}\right) \right] \right\}  . \end{align*}
</div>
<p>Therefore, taking the absolute values of both sides of the last formula, and using the hypothesis that \(f\in \operatorname {Lip}_{M}\left( \alpha ,[0,1]\right) \  \)in the result, we get</p>
<div class="displaymath" id="a0000000049">
  \begin{align*} & \left\vert L_{n,r}^{\beta }\left( f;y\right) -L_{n,r}^{\beta }\left( f;x\right) \right\vert \leq \\ & \leq \tfrac {1}{\left( 1+\left( n-r\right) \beta \right) ^{n-r-1}}\times \\ & \quad \times \sum \limits _{k=0}^{n-r}\sum \limits _{l=0}^{n-r-k}\tfrac {\left( n-r\right) !}{k!l!\left( n-r-k-l\right) !}x\left( x+k\beta \right) ^{k-1}\left( y-x\right) \left[ y-x+l\beta \right] ^{l-1} \\ & \quad \times \left( 1-y\right) \left[ 1-y+\left( n-r-k-l\right) \beta \right] ^{n-r-k-l-1} \\ & \quad \times \left\{  \left( 1-y\right) \left\vert f\left( \tfrac {k+l}{n}\right) -f\left( \tfrac {k}{n}\right) \right\vert +x\left\vert f\left( \tfrac {k+l+r}{n}\right) -f\left( \tfrac {k+r}{n}\right) \right\vert \right. \\ & \quad \left. +(y-x)\left\vert f\left( \tfrac {k+l+r}{n}\right) -f\left( \tfrac {k}{n}\right) \right\vert \right\}  \\ & \leq \tfrac {M}{\left( 1+\left( n-r\right) \beta \right) ^{n-r-1}}\times \\ & \quad \times \sum \limits _{k=0}^{n-r}\sum \limits _{l=0}^{n-r-k}\tfrac {\left( n-r\right) !}{k!l!\left( n-r-k-l\right) !}x\left( x+k\beta \right) ^{k-1}\left( y-x\right) \left[ y-x+l\beta \right] ^{l-1} \\ & \quad \times \left( 1-y\right) \left[ 1-y+\left( n-r-k-l\right) \beta \right] ^{n-r-k-l-1} \\ & \quad \times \left\{  \left( 1-\left( y-x\right) \right) \left( \tfrac {l}{n}\right) ^{\alpha }+(y-x)\left( \tfrac {l+r}{n}\right) ^{\alpha }\right\}  \end{align*}
</div>
<p>by the assumption that \(x\leq y.\  \)Changing the order of the summations, the above formula gives that</p>
<div class="displaymath" id="a0000000050">
  \begin{align*} & \left\vert L_{n,r}^{\beta }\left( f;y\right) -L_{n,r}^{\beta }\left( f;x\right) \right\vert \leq \\ & \leq \tfrac {M}{\left( 1\! +\! \left( n-r\right) \beta \right) ^{n-r-1}} \sum \limits _{l=0}^{n-r}\tbinom {n-r}{l}\left( y\! -\! x\right) \left[ y-x+l\beta \right] ^{l-1} \times \\ \end{align*}
</div>
<div class="displaymath" id="a0000000051">
  \begin{align*} & \quad \times \sum \limits _{k=0}^{n-r-l}\tbinom {n-r-l}{k}x\left( x+k\beta \right) ^{k-1}\left( 1-y\right) \left[ 1-y+\left( n-r-l-k\right) \beta \right] ^{n-r-l-k-1} \\ & \quad \times \left\{  \left( 1-\left( y-x\right) \right) \left( \tfrac {l}{n}\right) ^{\alpha }+(y-x)\left( \tfrac {l+r}{n}\right) ^{\alpha }\right\}  . \end{align*}
</div>
<p>By taking \(u=\) \(x,\) \(v=1-y\) and \(m=n-r-l\) in (<a href="#Jensen">3</a>), this formula can be simplified as </p>
<div class="displaymath" id="a0000000052">
  \begin{align*} & \left\vert L_{n,r}^{\beta }\left( f;y\right) -L_{n,r}^{\beta }\left( f;x\right) \right\vert \leq \\ & \leq \tfrac {M}{\left( 1+\left( n-r\right) \beta \right) ^{n-r-1}}\sum \limits _{l=0}^{n-r}\tbinom {n-r}{l}\left( y-x\right) \left[ y-x+l\beta \right] ^{l-1} \times \\ & \quad \times \left( 1-\left( y-x\right) \right) \left[ 1-\left( y-x\right) +\left( n-r-l\right) \beta \right] ^{n-r-l-1} \\ & \quad \times \left\{  \left( 1-\left( y-x\right) \right) \left( \tfrac {l}{n}\right) ^{\alpha }+(y-x)\left( \tfrac {l+r}{n}\right) ^{\alpha }\right\}  . \end{align*}
</div>
<p>Here, taking \(x_{1}=\frac{l}{n},\) \(x_{2}=\frac{l+r}{n},\  \)and regarding the nonnegative constants \(\alpha _{1},\   \alpha _{2}\  \)as\(\   \alpha _{1}=y-x\  \)and \(\alpha _{2}=1-\left( y-x\right) \  \)that satisfy\(\   \alpha _{1}+\alpha _{2}=1,\  \)and using the fact that \(g\left( t\right) =t^{\alpha },\) \(0{\lt}\alpha \leq 1,\  \)is concave, then the last formula reduces to </p>
<div class="displaymath" id="a0000000053">
  \begin{align*} & \left\vert L_{n,r}^{\beta }\left( f;y\right) -L_{n,r}^{\beta }\left( f;x\right) \right\vert \leq \\ & \leq \tfrac {M}{\left( 1+\left( n-r\right) \beta \right) ^{n-r-1}}\sum \limits _{l=0}^{n-r}\tbinom {n-r}{l}\left( y-x\right) \left[ y-x+l\beta \right] ^{l-1}\times \\ & \quad \times \left( 1-\left( y-x\right) \right) \left[ 1-\left( y-x\right) +\left( n-r-l\right) \beta \right] ^{n-r-l-1} \\ & \quad \times \left\{  \left( 1-\left( y-x\right) \right) \tfrac {l}{n}+(y-x)\left( \tfrac {l+r}{n}\right) \right\}  ^{\alpha } \\ & =M\sum \limits _{l=0}^{n-r}P_{n-r}^{\beta }\left( y-x\right) \left\{  \left( 1-\left( y-x\right) \right) \tfrac {l}{n}+(y-x)\left( \tfrac {l+r}{n}\right) \right\}  ^{\alpha }, \end{align*}
</div>
<p>where \(P_{n-r}^{\beta }\  \)is given by (<a href="#Pn,k,beta">6</a>)\(.\) Here, the case\( \alpha =1\  \)is obvious. For the case \(0{\lt}\alpha {\lt}1;\) application of Hölder’s inequality, with conjugate pairs \(p=\frac{1}{\alpha }\) and \(q=\frac{1}{1-\alpha },\  \)leads to</p>
<div class="displaymath" id="a0000000054">
  \begin{align*} & \left\vert L_{n,r}^{\beta }\left( f;y\right) -L_{n,r}^{\beta }\left( f;x\right) \right\vert \leq \\ & \leq M\left\{  \sum \limits _{l=0}^{n-r}P_{n-r}^{\beta }\left( y\! -\! x\right) \left[ \left( 1\! \! -\! \! \left( y\! \! -\! \! x\right) \right) \tfrac {l}{n}\! \! +\! \! (y\! -\! x)\left( \tfrac {l+r}{n}\right) \right] \right\}  ^{\alpha }\left\{  \sum \limits _{l=0}^{n-r}P_{n-r}^{\beta }\left( y\! -\! x\right) \right\}  ^{1-\alpha } \\ & =M\text{ }\left\{  L_{n,r}^{\beta }\left( e_{1};y-x\right) \right\}  ^{\alpha }\left\{  L_{n,r}^{\beta }\left( e_{0};y-x\right) \right\}  ^{1-\alpha } \\ & =M(y-x)^{\alpha }, \end{align*}
</div>
<p>by Lemma <a href="#momentsS_CS">3.1</a>, which completes this proof. <div class="proof_wrapper" id="a0000000055">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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</dd>
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  <dd><p><a href ="https://doi.org/10.1007/978-3-662-02888-9"> <i class="sc">R. DeVore, G.G. Lorentz</i>, <i class="it">Constructive Approximation</i>, Springer, Berlin, 1993. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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  <dd><p><a href =" https://doi.org/10.1007/978-3-0348-6308-7_23"> <i class="sc">D.D. Stancu</i>, <i class="it">Quadrature formulas constructed by using certain linear positive operators,</i> Numerical Integration (Proc. Conf., Oberwolfach, 1981), ISNM 57 (1982), pp.&#160;241–251, Birkhäuser Verlag, Basel. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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  <dd><p><a href ="https://doi.org/10.1007/bf02575593"> <i class="sc">D.D. Stancu</i>, <i class="it">Approximation of functions by means of a new generalized Bernstein operator, </i>Calcolo, <b class="bfseries">20</b> (1983) no. 2, pp.&#160;211–229. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
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</dd>
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</dd>
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</dd>
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</dd>
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  <dd><p><a href ="https://doi.org/10.1016/s0096-3003(02)00088-7"> <i class="sc">R. Yang, J. Xiong, F. Cao</i>, <i class="it">Multivariate Stancu operators defined on a simplex,</i> Appl. Math. Comput., <b class="bfseries">138</b> (2003), pp.&#160;189–198. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
</dl>


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