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<h1>Durrmeyer-Stancu type operators</h1>
<p class="authors">
<span class="author">Ovidiu T. Pop\(^\ast \) Dan Bărbosu\(^\S \)</span>
</p>
<p class="date">August 05, 2008</p>
</div>
<p>\(^\ast \)National College “Mihai Eminescu”, 5 Mihai Eminescu Street, 440014 Satu Mare,<br />Romania, e-mail: <span class="tt">ovidiutiberiu@yahoo.com</span> </p>
<p>\(^\S \)North University of Baia Mare, Dept. of Math. and Comp. Science, Victoriei 76, 43012 Baia Mare, Romania, e-mail: <span class="tt">danbarbosu@yahoo.com</span> </p>

<div class="abstract"><p> Considering two given real parameters \(\alpha , \beta \) satisfying the conditions \(0\leq \alpha \leq \beta \). D. D. Stancu [7] constructed and studied the linear positive operators \(P^{(\alpha ,\beta )}_m:C([0,1])\to C([0,1])\), defined for any \(f\in C([0,1])\) and any positive integer \(m\) by (<a href="#1.1">1</a>). In this paper we are dealing with the Durrmeyer form of Stancu’s operators. Some approximation properties of these Durrmeyer-Stancu operators are established. As a particular case, we retrieve approximation properties for the classical Durrmeyer operators [5]. </p>
<p><b class="bf">MSC.</b> 41A10, 41A25, 41A36, 41A63. </p>
<p><b class="bf">Keywords.</b> Linear positive operators, Durrmeyer operators, first order modulus of smoothness, Shisha-Mond theorem. </p>
</div>
<h1 id="a0000000002">1 Preliminaries</h1>
<p> Let \(\alpha , \beta \) be real parameters satisfying the conditions \(0\leq \alpha \leq \beta \).<br />In 1969 D. D. Stancu [7] constructed and studied the linear positive operators \(P^{(\alpha ,\beta )}_m:C([0,1])\to C([0,1])\) defined for any \(f\in C([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\) by: </p>
<div class="equation" id="1.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.1} \left(p^{(\alpha ,\beta )}_m f\right)(x)=\sum ^m_{k=0} p_{m,k}(x) f\left(\tfrac {k+\alpha }{m+\beta }\right) \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> where </p>
<div class="equation" id="1.2">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.2} p_{m,k}(x)=\tbinom {m}{k} x^k(1-x)^{m-k} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> are the fundamental Bernstein polynomials [4]. </p>
<p>In the monograph by F. Altomare and M. Campiti [2], the operators (<a href="#1.1">1</a>) are called “the Bernstein-Stancu operators”. </p>
<p>In 1967, J. L. Durrmeyer [5] introduced the operators \(D_m:L_1([0,1])\to C([0,1])\), defined for any \(f\in C([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\) by: </p>
<div class="equation" id="1.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{1.3} (D_m f)(x)=(m+1)\sum ^m_{k=0}p_{m,k}(x)\int ^1_0 p_{mk}(t) f(t){\rm d}t. \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<p> Note that the operators (<a href="#1.3">3</a>) are known in the mathematical literature as the “Durrmeyer operators”. Following the Durrmeyer ideas, the second author introduced and studied Durrmeyer-Schurer operators [3]. </p>
<p>The aim of the present paper is to construct the Durrmeyer type operators associated to the operators (<a href="#1.1">1</a>), denoted Durrmeyer-Stancu operators and to establish some of them approximation properties. </p>
<p>In Section 2 we recall the well-known theorem due to O. Shisha and B. Mond [6] and some properties of Durrmeyer operators [5]. We also introduce here the Durrmeyer-Stancu operators, denoted \(D_m^{(\alpha ,\beta )}\) and we compute the images of test functions by these operators. </p>
<p>Section 3 contains the main results of the paper which are: a convergence theorem for the sequence \(\big\{ D_m^{(\alpha ,\beta )} f\big\} _{m\in \mathbb N}\); the approximation order of \(f\) by \(D_m^{(\alpha ,\beta )} f\), under different assumption on the approximated function \(f\).<br />As particular case, we get the classical Durrmeyer operator (<a href="#1.3">3</a>). </p>
<h1 id="a0000000003">2 Auxiliary results</h1>
<p>As usually, \(\omega _1(f;\delta )\) denotes the first order modulus of smoothness for the function \(f\) and \(e_j(x)=x^j\)  \((j=0,1,\dots )\) are the test functions. </p>
<p>First, we recall the following result, due to O. Shisha and B. Mond [6]. <div class="theorem_thmwrapper " id="a0000000004">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(I\) be a non-empty interval of real axis, let \(L:C_B(I)\to B(I)\) be a linear and positive operator and, for \(x\in I\), let \(\varphi _x:I\to \mathbb R\) be defined by \(\varphi _x(t)=|t-x|\), for any \(t\in I\). </p>
<ul class="itemize">
  <li><p>Let \(f\in C_B(I)\) be given; for any \(\delta {\gt}0\) and any \(x\in I\) the following </p>
<div class="displaymath" id="2.1">
  \begin{align} \label{2.1} |(Lf)(x)-f(x)|& \leq |f(x)||(Le_0)(x)-1|+\\ & \quad +\left\{ (Le_0)(x)+\delta ^{-1}\sqrt{(Le_0)(x)(L\varphi ^2_x)(x)} \right\} \omega _1(f;\delta )\nonumber \end{align}
</div>
<p> holds. </p>
</li>
  <li><p>Let \(f\) be a given differentiable function such that \(f'\in C_B(I)\); for any \(\delta {\gt}0\) and any \(x\in I\), the following </p>
<div class="displaymath" id="2.2">
  \begin{align} \label{2.2} |(Lf)(x)-f(x)|& \leq |f(x)||(Le_0)(x)-1|+\\ & \quad +|f’(x)||(Le_1)(x)-x(Le_0)(x)|+\nonumber \\ & \quad +\sqrt{(L\varphi ^2_x)(x)}\left\{ \sqrt{Le_0)(x)}+ \delta ^{-1}\sqrt{(L\varphi ^2_x)(x)}\right\}  \omega _1(f’;\delta )\nonumber \end{align}
</div>
<p> holds. </p>
</li>
</ul>

  </div>
</div> Next, let us to recall some properties of Durrmeyer operators. <div class="lemma_thmwrapper " id="a0000000005">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2</span>
  </div>
  <div class="lemma_thmcontent">
  <p><em>[5]</em>. For any positive integer \(m\) and any \(x\in [0,1]\), the Durrmeyer operators \((\ref{1.3})\) verify: </p>
<div class="displaymath" id="2.3">
  \begin{align} \label{2.3} & (D_m e_0)(x)=1;\\ & (D_m e_1)(x)=\tfrac {mx+1}{m+2}\, ;\label{2.4}\\ & (D_m e_2)(x)=\tfrac {m(m-1)x^2+4mx+2}{(m+2)(m+3)}\, ;\label{2.5}\\ & (D_m\varphi ^2_x)(x)=\tfrac {2(m-3)x(1-x)+2}{(m+2)(m+3)}\, .\label{2.6} \end{align}
</div>

  </div>
</div> <div class="definition_thmwrapper " id="a0000000006">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">3</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(\alpha , \beta \) be real parameters satisfying the conditions \(0\leq \alpha \leq \beta \). The Durrmeyer-Stancu operators \(D_m^{(\alpha ,\beta )}:L_1([0,1])\to C([0,1])\) are defined for any \(f\in L_1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\) by: </p>
<div class="equation" id="2.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.7} \left(D_m^{(\alpha ,\beta )} f\right)(x)=(m+1)\sum ^m_{k=0} p_{m,k}(x)\int ^1_0 p_{m,k}(t) f\left(\tfrac {mt+\alpha }{m+\beta }\right){\rm d}t. \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>

  </div>
</div> <div class="lemma_thmwrapper " id="a0000000007">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">4</span>
  </div>
  <div class="lemma_thmcontent">
  <p>The operators \((\ref{2.7})\) are linear and positive. </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000008">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Directly, from Definition 3. <div class="lemma_thmwrapper " id="a0000000009">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">5</span>
  </div>
  <div class="lemma_thmcontent">
  <p>For any \(x\in [0,1]\) and any positive integer \(m\) the Durrmeyer-Stancu operators \((\ref{2.7})\) verify: </p>
<div class="equation" id="2.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.8} \left(D_m^{(\alpha ,\beta )}e_0\right)(x)=1; \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<div class="equation" id="2.9">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.9} \left(D_m^{(\alpha ,\beta )} e_1\right)(x)=\tfrac {m^2}{(m+\beta )(m+2)}\, x+ \tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)}\, ; \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<div class="displaymath" id="2.10">
  \begin{align} \label{2.10} \left(D_m^{(\alpha ,\beta )} e_2\right)(x)& = \tfrac {m^3(m-1)}{(m+\beta )^2(m+2)(m+3)}\, x^2\\ & \quad +\tfrac {4m^3+2\alpha m^2(m+3)}{(m+\beta )^2(m+2)(m+3)}\, x\nonumber \\ & \quad +\tfrac {2m^2+2\alpha m(m+3)+\alpha ^2(m+2)(m+3)} {(m+\beta )^2(m+2)(m+1)}\, ;\nonumber \end{align}
</div>
<div class="displaymath" id="2.11">
  \begin{align} \label{2.11} \left(D_m^{(\alpha ,\beta )}\varphi ^2_x\right)(x)& = \left(\tfrac {m}{m+\beta }\right)^2 \tfrac {2(m-3)x(1-x)+2}{(m+2)(m+3)}+\\ & \quad +\! \tfrac {\! \left\{ \! \beta ^2(m\! +\! 2)\! +\! 4\beta m\! \right\} x^2\! -\! 2 \left\{ \! \alpha \beta (m\! +\! 2)\! +\! \beta m\! +\! 2\alpha m\! \right\} x\! + \! \alpha ^2(m\! +\! 2)\! +\! 2m\alpha } {(m+\beta )^2 (m+2)}.\nonumber \end{align}
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000010">
  <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 observe the following identities: </p>
<div class="displaymath" id="a0000000011">
  \begin{align*}  \left(D_m^{(\alpha ,\beta )}e_0\right)(x)& =\left(D_m e_0\right)(x)\\ \left(D_m^{(\alpha ,\beta )}e_1\right)(x)& =\tfrac {m}{m+\beta }\,  (D_m e_1)(x)+\tfrac {\alpha }{m+\beta }\, (D_m e_0)(x)\\ \left(D_m^{(\alpha ,\beta )}e_2\right)(x)& = \left(\tfrac {m}{m+\beta }\right)^2(D_m e_2)(x)+ \tfrac {2\alpha m}{(m+\beta )^2}\, (D_m e_1)(x)+\\ & \quad +\tfrac {\alpha ^2}{(m+\beta )^2}\, (D_m e_0)(x)\end{align*}
</div>
<div class="displaymath" id="a0000000012">
  \begin{align*}  \left(D_m^{(\alpha ,\beta )}\varphi _x^2\right)(x)& = \left(D_m^{(\alpha ,\beta )}e_2\right)(x)-2x\left(D_m^{(\alpha ,\beta )}e_1\right)(x) +x^2\left(D_m^{(\alpha ,\beta )}e_0\right)(x). \end{align*}
</div>
<p> Next, one applies Lemma 2. <div class="lemma_thmwrapper " id="a0000000013">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">6</span>
  </div>
  <div class="lemma_thmcontent">
  <p>For any positive integer \(m\) and any \(x\in [0,1]\) the following </p>
<div class="equation" id="2.12">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.12} \delta _m^{(\alpha ,\beta )}(x)\leq \delta ^{(\alpha ,\beta )}_m \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> holds, where </p>
<div class="equation" id="2.13">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.13} \delta _m^{(\alpha ,\beta )}(x)=\sqrt{\left(D_m^{(\alpha ,\beta )} \varphi ^2_m\right)(x)} \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p> and </p>
<div class="equation" id="2.14">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.14} \delta _m^{(\alpha ,\beta )}= \sqrt{\! \left(\! \tfrac {m}{m\! +\! \beta }\! \right)^2 \tfrac {m\! +\! 1}{2(m\! +\! 2)(m\! +\! 3)}\! +\! \max \left\{ \! \tfrac {\alpha ^2(m\! +\! 2)\! +\! 2m\alpha } {(m\! +\! \beta )^2(m\! +\! 2)}\, ,\tfrac {(m\! +\! 2)(\alpha \! -\! \beta )^2\! -\! 2m(\alpha \! -\! \beta )} {(m+\beta )^2(m+2)}\right\} \! } .\end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>

  </div>
</div> <div class="proof_wrapper" id="a0000000014">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Because \(x(1-x)\leq \tfrac {1}{4}\) for any \(x\in [0,1]\), we get </p>
<div class="equation" id="2.15">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.15} \tfrac {2(m-3)x(1-x)+2}{(m+2)(m+3)}\leq \tfrac {m+1}{2(m+2)(m+3)} \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> for any positive integer \(m\). </p>
<p>For any positive integer \(m\), let us to introduce the following notations: </p>
<div class="displaymath" id="a0000000015">
  \begin{align*}  \gamma ^{(\alpha ,\beta )}_m & =\max \left\{ \tfrac {\alpha ^2(m+2)+2m\alpha } {(m+\beta )^2(m+2)}\, ,\tfrac {(m+2)(\alpha -\beta )^2-2m(\alpha -\beta )} {(m+\beta )^2(m+2)}\right\} ;\\ a_m & =\tfrac {\beta ^2(m+2)+4\beta m}{(m+\beta )^2(m+2)}\, ;\\ b_m& =-2\tfrac {\alpha \beta (m+2)+\beta m+2\alpha m} {(m+\beta )^2(m+2)}\, ;\\ c_m & =\tfrac {\alpha ^2(m+2)+2m\alpha }{(m+\beta )^2(m+2)}\, . \end{align*}
</div>
<p> Let \(f_m[0,1]\to \mathbb R\) be defined for any \(x\in [0,1]\) by: </p>
<div class="displaymath" id="a0000000016">
  \[  f_m(x)=a_m x^2+b_m x+c_m.  \]
</div>
<p>If \(\beta =0\), follows \(\alpha =0\) and \(f_m(x)=0\), for any \(x\in [0,1]\). </p>
<p>If \(\beta \neq 0\) the function \(f\) is a polynomial function of second degree and because \(a_m{\gt}0\) follows that \(f_m\) has a maximum value in the point </p>
<div class="displaymath" id="a0000000017">
  \[  x_m=-\tfrac {b_m}{2a_m}= \tfrac {\alpha \beta (m+2)+\beta m+2\alpha m} {\beta ^2(m+2)+4\beta m}\, .  \]
</div>
<p> Because \(\alpha \leq \beta \), we get </p>
<div class="displaymath" id="a0000000018">
  \[  x_m\leq \tfrac {\beta ^2(m+2)+3\beta m} {\beta ^2(m+2)+4\beta m}{\lt}1.  \]
</div>
<p> Taking the above inequality into account, yields: </p>
<div class="displaymath" id="a0000000019">
  \[  f_m(x)\leq \max \left\{ f_m(0), f_m(1)\right\} ,  \]
</div>
<p> for any \(x\in [0,1]\). Follows that for any \(x\in [0,1]\) the following </p>
<div class="equation" id="2.16">
<p>
  <div class="equation_content">
    \begin{equation} \label{2.16} f_m(x)\leq \gamma ^{(\alpha ,\beta )}_m. \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p> Applying (<a href="#2.15">18</a>) and (<a href="#2.16">19</a>) we get (<a href="#2.12">15</a>). </p>
<h1 id="a0000000020">3 Main results</h1>
<p> <div class="theorem_thmwrapper " id="a0000000021">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>For any \(f\in L_1([0,1])\) the sequence \(\big\{ D^{(\alpha ,\beta )}_m f\big\} \) converges to \(f\), uniformly on \([0,1]\). </p>

  </div>
</div> <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>Lemma 5 (identity (<a href="#2.11">14</a>)) follows \(\lim \limits _{m\to \infty }\big(D_m^{(\alpha ,\beta )}\varphi ^2_x\big)(x)=0\), uniformly on \([0,1]\). Then one applies the well known Bohman-Korovkin theorem ([1] or [8]). </p>
<p><div class="theorem_thmwrapper " id="a0000000023">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">8</span>
  </div>
  <div class="theorem_thmcontent">
  <ul class="itemize">
  <li><p>For any \(f\in L_1([0,1])\), any \(x\in [0,1]\), any \(\delta {\gt}0\) and any positive integer \(m\), the following </p>
<div class="equation" id="3.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.1} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right|\leq \left(1+\tfrac {1}{\delta }\, \delta _m^{(\alpha ,\beta )}(x)\right) \omega _1 (f;\delta ) \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p> holds. </p>
</li>
  <li><p>For any \(f\in C^1([0,1])\), any \(x\in [0,1]\), any \(\delta {\gt}0\) and any positive integer \(m\), the following </p>
<div class="displaymath" id="3.2">
  \begin{align} \label{3.2} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right|& \leq |f’(x)|\bigg|-\tfrac {(\beta +2)m+2\beta }{(m+\beta )(m+2)}\, x +\tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)}\bigg|+\\ & \quad +\delta _m^{(\alpha ,\beta )}(x)\left(1+\tfrac {1}{\delta }\,  \delta _m^{(\alpha ,\beta )}(x)\right)\omega _1(f’;\delta )\nonumber \end{align}
</div>
<p> holds. </p>
</li>
</ul>
<p> In \((\ref{3.1})\) and \((\ref{3.2})\) \(\delta _m^{(\alpha ,\beta )}(x)\) is defined at \((\ref{2.13})\). </p>

  </div>
</div> <div class="proof_wrapper" id="a0000000024">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>One applies Theorem 1 and Lemma 5. </p>
<p><div class="theorem_thmwrapper " id="a0000000025">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">9</span>
  </div>
  <div class="theorem_thmcontent">
  <ul class="itemize">
  <li><p>For any \(f\in L_1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\), the following </p>
<div class="equation" id="3.3">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.3} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right| \leq 2\omega _1\left(f; \delta _m^{(\alpha ,\beta )}(x)\right) \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p> holds. </p>
</li>
  <li><p>For any \(f\in C^1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\) the following </p>
<div class="displaymath" id="3.4">
  \begin{align} \label{3.4} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right|& \leq |f’(x)|\, \bigg|\tfrac {(\beta +2)m+2\beta }{(m+\beta )(m+2)}\, x+ \tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)}\bigg|+\\ & \quad +2\delta _m^{(\alpha ,\beta )}(x)\omega _1\left(f’; \delta _m^{(\alpha ,\beta )} (x)\right).\nonumber \end{align}
</div>
</li>
</ul>

  </div>
</div> <div class="proof_wrapper" id="a0000000026">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>In Theorem 8 we choose \(\delta =\delta _{m,p}^{(\alpha ,\beta )}(x)\). <div class="remark_thmwrapper " id="a0000000027">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">10</span>
  </div>
  <div class="remark_thmcontent">
  <p>For any positive integer \(m\), let \(g_m:[0,1]\to \mathbb R\) be defined by </p>
<div class="displaymath" id="a0000000028">
  \[  g_m(x)=-\tfrac {(\beta +2)m+2\beta }{(m+\beta )(m+2)}\, x+ \tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)}  \]
</div>
<p> for any \(x\in [0,1]\). It is immediately that </p>
<div class="equation" id="3.5">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.5} |g_m(x)|\leq \eta _m^{(\alpha ,\beta )} \end{equation}
  </div>
  <span class="equation_label">24</span>
</p>
</div>
<p> for any \(x\in [0,1]\), where </p>
<div class="displaymath" id="3.6">
  \begin{align} \label{3.6} \eta _m^{(\alpha ,\beta )} & =\max \left\{ |g_m(0)|, g_m(1)\right\} =\\ & =\max \left\{ \tfrac {(\alpha +1)m+2\alpha }{(m+\beta )(m+2)}\, , \tfrac {|m(\alpha -\beta -1)+2(\alpha -\beta )|} {(m+\beta )(m+2)}\right\} .\nonumber \end{align}
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> Applying then Theorem 9, Lemma 6 and (<a href="#3.5">24</a>), we get: <div class="corollary_thmwrapper " id="a0000000029">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">11</span>
  </div>
  <div class="corollary_thmcontent">
  <p>\((i)\) For any \(f\in L_1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\), the following </p>
<div class="equation" id="3.7">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.7} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right) \leq 2\omega _1\left(f; \delta _m^{(\alpha ,\beta )}\right) \end{equation}
  </div>
  <span class="equation_label">26</span>
</p>
</div>
<p> holds. </p>
<p>\((ii)\) For any \(f\in C^1([0,1])\), any \(x\in [0,1]\) and any positive integer \(m\), the following </p>
<div class="equation" id="3.8">
<p>
  <div class="equation_content">
    \begin{equation} \label{3.8} \left|\left(D_m^{(\alpha ,\beta )} f\right)(x)-f(x)\right|\leq M_1\eta _m^{(\alpha ,\beta )}+2\delta _m^{(\alpha ,\beta )} \omega _1\left(f'; \delta _m^{(\alpha ,\beta )}\right) \end{equation}
  </div>
  <span class="equation_label">27</span>
</p>
</div>
<p> where \(M_1=\max \limits _{x\in [0,1]}|f'(x)|\). </p>

  </div>
</div> <div class="remark_thmwrapper " id="a0000000030">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">12</span>
  </div>
  <div class="remark_thmcontent">
  <p>For any \(\alpha =\beta =0\), the operators \(D_m^{(0,0)}\) are the Durrmeyer operators.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="agra">1</a></dt>
  <dd><p><i class="sc">O. Agratini</i>, <em>Aproximare prin operatori liniari</em>, Presa Universitară Clujeană, Cluj-Napoca, 2000 (in Romanian). </p>
</dd>
  <dt><a name="alto">2</a></dt>
  <dd><p><i class="sc">F. Altomare</i> and <i class="sc">M. Campiti</i>, <em>Korovkin-type Approximation Theory and its Applications</em>, de Gruyter, Series Studies in Mathematics, <b class="bfseries">17</b>, Walter de Gruyter&amp;Co, Berlin, New-York, 2000. </p>
</dd>
  <dt><a name="barb">3</a></dt>
  <dd><p><i class="sc">D. Bărbosu</i>, <em>Durrmeyer-Schurer type operators</em>, Facta Univ. (Nis̃), Ser. Math-Inform., <b class="bfseries">19</b>, pp.&#160;65–72, 2004. </p>
</dd>
  <dt><a name="bern">4</a></dt>
  <dd><p><i class="sc">S.N. Bernstein</i>, <em>Demonstration du théorème de Weierstrass fondeé sur le calcul des probabilités</em>, Commun. Soc. Math. Kharkhov, <b class="bfseries">13</b>(<b class="bfseries">2</b>), pp.&#160;1–2, 1912–1913. </p>
</dd>
  <dt><a name="durr">5</a></dt>
  <dd><p><i class="sc">J.L. Durrmeyer</i>, <em>Une formule d’inversion de transformée de Laplace: Application à la theorie des moments</em>, Thèse de 3e cycle, Faculté de Sciences de l’Université de Paris, 1967. </p>
</dd>
  <dt><a name="shisa">6</a></dt>
  <dd><p><i class="sc">O. Shisha</i> and <i class="sc">B. Mond</i>, <em>The degree of convergence of linear positive operators</em>, Proc. Nat. Acad. Sci. U.S.A, <b class="bfseries">60</b>, pp.&#160;1196–2000, 1968. </p>
</dd>
  <dt><a name="stan">7</a></dt>
  <dd><p><i class="sc">D.D. Stancu</i>, <em>Asupra unei generalizări a polinoamelor lui Bernstein</em>, Studia Univ. “Babeş-Bolyai”, <b class="bfseries">14</b>(<b class="bfseries">2</b>), pp.&#160;31–45, 1969 (in Romanian). </p>
</dd>
  <dt><a name="stan">7</a></dt>
  <dd><p><i class="sc">D.D. Stancu</i>, <i class="sc">Gh. Coman</i>, <i class="sc">O. Agratini</i>, <i class="sc">R. Trîmbiţaş</i>, <em>Analiză Numerică şi Teoria Aproximării</em>, <b class="bfseries">I</b>, Presa Universitară Clujeană, Cluj-Napoca, 2001 (in Romanian). </p>
</dd>
</dl>


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