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Approximation by modified Kantorovich–Stancu operators

Journal of Inequalities and Applications20182018:346

https://doi.org/10.1186/s13660-018-1939-9

  • Received: 11 April 2018
  • Accepted: 6 December 2018
  • Published:

Abstract

In the present paper, we study a new kind of Kantorovich–Stancu type operators. For this modified form, we discuss a uniform convergence estimate. Some Voronovskaja-type theorems are given.

Keywords

  • Stancu operators
  • Kantorovich operators
  • Stancu–Kantorovich operators
  • Voronovskaja-type theorem

MSC

  • 41A25
  • 41A36

1 Introduction

Let \(0 \le \alpha \le \beta\) and \(m \in N\). In [15], D.D. Stancu introduced the linear positive operators
$$P_{m}^{ ( \alpha,\beta )}:C \bigl( [0,1] \bigr) \to C \bigl( [0,1] \bigr) $$
defined by
$$ P_{m}^{ ( \alpha,\beta )} ( f;x ) = \sum_{k = 0}^{m}p_{m,k} ( x ) f \biggl( \frac{k + \alpha}{m + \beta} \biggr), $$
(1.1)
where
$$p_{m,k} ( x ) = \begin{pmatrix} m \\ k \end{pmatrix} x^{k} ( 1 - x )^{m - k} $$
are the fundamental Bernstein polynomials [3].
When \(\alpha = \beta = 0\),
$$P_{m}^{ ( 0,0 )} ( f;x ) = B_{m} ( f;x ) $$
is the classical Bernstein operator.
L.V. Kantorovich [8] introduced the linear positive operators
$$K_{m}:L_{1} \bigl( [0,1] \bigr) \to C \bigl( [0,1] \bigr) $$
defined for any nonnegative integer m by
$$ K_{m} ( f;x ) = ( m + 1 )\sum_{k = 0}^{m} p_{m,k} ( x ) \int_{\frac{k}{m + 1}}^{\frac{k + 1}{m + 1}} f ( s )\,ds. $$
(1.2)
By combining (1.1) and (1.2), D. Bărbosu [2] introduced
$$K_{m}^{ ( \alpha,\beta )}:L_{1} \bigl( [0,1] \bigr) \to C \bigl( [0,1] \bigr) $$
defined for any \(m \in N\) by
$$ K_{m}^{ ( \alpha,\beta )} ( f;x ) = ( m + \beta + 1 )\sum _{k = 0}^{m} p_{m,k} ( x ) \int_{\frac{k + \alpha}{m + \beta + 1}}^{\frac{k + \alpha + 1}{m + \beta + 1}} f ( s )\,ds. $$
(1.3)
\(K_{m}^{ ( \alpha,\beta )}\) are linear positive operators called Kantorovich–Stancu operators.

In recent years, Bernstein–Kantorovich–Stancu operators have been modified and studied by many mathematicians. For instance, in [4] Cai et al. defined a new type λ-Bernstein operators, and a Kantorovich variant of the modified Bernstein operators was introduced and studied in [7]. In the last three years, Mursaleen et al. investigated several approximation properties for a Kantorovich type generalization of q-Bernstein–Stancu operators in [14], applied (\(p,q\))-calculus in approximation theory, and constructed the (\(p,q\))-analogue of Bernstein operators [12], (\(p,q\))-Bernstein–Kantorovich operators [13], and a Kantorovich variant of (\(p,q\))-Szász–Mirakjan operators [11]. Also, in [1] Ansari and Karaisa introduced and studied Chlodowsky variant of (\(p,q\))-Bernstein operators.

H. Khosravian-Arab, M. Delghan, and M.R. Eslahchi introduced in [9] the following operators:
$$ B_{m}^{M,1} ( f;x ) = \sum_{k = 0}^{m} p_{m,k}^{M,1} ( x )f \biggl( \frac{k}{m} \biggr), $$
(1.4)
where
$$ p_{m,k}^{M,1} ( x ) = a ( x;m )p_{m - 1,k} ( x ) + a ( 1 - x;m )p_{m - 1,k - 1} ( x ),\quad x \in [0,1] $$
(1.5)
and
$$ a ( x;m ) = a_{1} ( m )x + a_{0} ( m ),\quad m = 0,1, \ldots . $$
(1.6)
Here, \(a_{0} ( m )\) and \(a_{1} ( m )\) are two unknown sequences which are determined in an appropriate way. Note that, for \(a_{0} ( m ) = 1\) and \(a_{1} ( m ) = - 1\), (1.5) becomes the well-known identity for the fundamental Bernstein polynomials
$$p_{m,k} ( x ) = ( 1 - x )p_{m - 1,k} ( x ) + xp_{m - 1,k - 1} ( x ),\quad 0 < k < m. $$
From (1.5), the operators (1.4) become
$$\begin{aligned} B_{m}^{M,1} ( f;x ) =& a ( x;m )\sum _{k = 0}^{m} p_{m - 1,k} ( x )f \biggl( \frac{k}{m} \biggr) + a ( 1 - x;m )\sum_{k = 0}^{m} p_{m - 1,k - 1} ( x )f \biggl( \frac{k}{m} \biggr) \\ =& a ( x;m )P_{m - 1}^{ ( 0,1 )} ( f;x ) + a ( 1 - x;m )P_{m - 1}^{ ( 1,1 )} ( f;x ). \end{aligned}$$
We try to extend some results to the Kantorovich–Stancu operators considering the operators denoted by
$$ \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) = ( m + \beta + 1 )\sum _{k = 0}^{m} p_{m,k}^{M,1} ( x ) \int_{\frac{k + \alpha}{m + \beta + 1}}^{\frac{k + \alpha + 1}{m + \beta + 1}} f ( s )\,ds,\quad m \in N, x \in [0,1]. $$
(1.7)

2 Auxiliary results

Lemma 2.1

For \(p \in N^{ *}\), we have
$$\begin{aligned} &\mathrm{(i)}\quad P_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{p};x \bigr) = \sum_{i = 0}^{p} \begin{pmatrix} p \\ i \end{pmatrix} \frac{1}{ ( m + \beta )^{p - i}}B_{m} \bigl( ( \alpha - t\beta )^{p - i} ( t - x )^{i};x \bigr); \\ &\mathrm{(ii)}\quad K_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{p};x \bigr) = \frac{1}{p + 1}\sum_{i = 1}^{p + 1} \begin{pmatrix} p + 1 \\ i \end{pmatrix} \frac{1}{ ( m + \beta + 1 )^{i - 1}}P_{m}^{ ( \alpha,\beta + 1 )} \bigl( ( t - x )^{p + 1 - i};x \bigr). \end{aligned}$$

Proof

(i)
$$\begin{aligned} P_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{p};x \bigr) &= \sum _{k = 0}^{m} p_{m,k} ( x ) \biggl( \frac{k + \alpha}{m + \beta} - x \biggr)^{p} = \sum_{k = 0}^{m} p_{m,k} ( x ) \biggl( \frac{k}{m} - x + \frac{m\alpha - k\beta}{m ( m + \beta )} \biggr)^{p} \\ &= \sum_{k = 0}^{m} p_{m,k} ( x ) \left ( \sum_{i = 0}^{p} \begin{pmatrix} p \\ i \end{pmatrix} \biggl( \frac{k}{m} - x \biggr)^{i} \biggl( \frac{m\alpha - k\beta}{m ( m + \beta )} \biggr)^{p - i} \right ) \\ &= \sum_{k = 0}^{m} p_{m,k} ( x ) \left ( \sum_{i = 0}^{p} \begin{pmatrix} p \\ i \end{pmatrix} \frac{1}{ ( m + \beta )^{p - i}} \biggl( \alpha - \frac{k}{m}\beta \biggr)^{p - i} \biggl( \frac{k}{m} - x \biggr)^{i} \right ) \\ &= \sum_{i = 0}^{p} \begin{pmatrix} p \\ i \end{pmatrix} \frac{1}{ ( m + \beta )^{p - i}} \Biggl( \sum_{k = 0}^{m} p_{m,k} ( x ) \biggl( \alpha - \frac{k}{m}\beta \biggr)^{p - i} \biggl( \frac{k}{m} - x \biggr)^{i} \Biggr) \\ &= \sum_{i = 0}^{p} \begin{pmatrix} p \\ i \end{pmatrix} \frac{1}{ ( m + \beta )^{p - i}}B_{m} \bigl( ( \alpha - t\beta )^{p - i} ( t - x )^{i};x \bigr). \end{aligned}$$
(ii) We have that
$$\begin{aligned} &K_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{p};x \bigr)\\ &\quad = \frac{m + \beta + 1}{p + 1}\sum_{k = 0}^{m} p_{m,k} ( x ) \biggl[ \biggl( \frac{k + \alpha}{m + \beta + 1} - x + \frac{1}{m + \beta + 1} \biggr)^{p + 1} - \biggl( \frac{k + \alpha}{m + \beta + 1} - x \biggr)^{p + 1} \biggr] \\ &\quad = \frac{m + \beta + 1}{p + 1}\sum_{k = 0}^{m} p_{m,k} ( x )\left ( \sum_{i = 1}^{p + 1} \begin{pmatrix} p + 1 \\ i \end{pmatrix} \biggl( \frac{1}{m + \beta + 1} \biggr)^{i} \biggl( \frac{k + \alpha}{m + \beta + 1} - x \biggr)^{p + 1 - i} \right ) \\ &\quad = \frac{1}{p + 1}\sum_{i = 1}^{p + 1} \begin{pmatrix} p + 1 \\ i \end{pmatrix}\frac{1}{ ( m + \beta + 1 )^{i - 1}} \Biggl( \sum_{k = 0}^{m} p_{m,k} ( x ) \biggl( \frac{k + \alpha}{m + \beta + 1} - x \biggr)^{p + 1 - i} \Biggr) \\ &\quad = \frac{1}{p + 1}\sum_{i = 1}^{p + 1} \begin{pmatrix} p + 1 \\ i \end{pmatrix}\frac{1}{ ( m + \beta + 1 )^{i - 1}} P_{m}^{ ( \alpha,\beta + 1 )} \bigl( ( t - x )^{p + 1 - i};x \bigr). \end{aligned}$$
 □

Corollary 2.2

For any \(p \in N^{ *}\), there exists a constant \(C(p)\), independent of m and x, such that
$$ \bigl\vert K_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{p};x \bigr) \bigr\vert \le C ( p ) \biggl( \frac{x ( 1 - x )}{m} \biggr)^{\frac{p}{2}} + O \biggl( \frac{1}{m^{p}} \biggr) $$
(2.1)
for every \(x \in [0,1]\).

Proof

First we have
$$ \bigl\vert P_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{p};x \bigr) \bigr\vert \le \sum_{i = 0}^{p} \begin{pmatrix} p \\ i \end{pmatrix} \biggl( \frac{M}{m + \beta} \biggr)^{p - i}B_{m} \bigl( \vert t - x \vert ^{i};x \bigr), $$
(2.2)
where \(M = \max \{ \alpha,\beta - \alpha \}\) for \(x \in [0,1 ]\).
The following inequality
$$ B_{m} \bigl( \vert t - x \vert ^{i};x \bigr) \le c ( i )\sqrt{ \biggl( \frac{X}{m} \biggr)^{i}}, $$
(2.3)
where \(c ( i )\) is a constant independent of m, can be found in [16] for \(mX \ge 1\), \(X = x ( 1 - x )\) and in [5] for \(mX < 1\).
Taking \(c ( p ) = \max_{i = \overline{0,p}}c ( i )\) in (2.3), by (2.2) it follows
$$\begin{aligned} \bigl\vert P_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{p};x \bigr) \bigr\vert \le& c ( p )\sum_{i = 0}^{p} \begin{pmatrix} p \\ i \end{pmatrix} \sqrt{ \biggl( \frac{x ( 1 - x )}{m} \biggr)^{i}} \biggl( \frac{M}{m + \beta} \biggr)^{p - i} \\ \le& c ( p ) \biggl( \sqrt{ \biggl( \frac{x ( 1 - x )}{m} \biggr)} + \frac{M}{m + \beta} \biggr)^{p} \\ \le& 2^{p - 1}c ( p ) \biggl( \sqrt{ \biggl( \frac{x ( 1 - x )}{m} \biggr)}^{p} + \biggl( \frac{M}{m + \beta} \biggr)^{p} \biggr) \\ \le& C ( p ) \biggl( \frac{x ( 1 - x )}{m} \biggr)^{\frac{p}{2}} + O \biggl( \frac{1}{m^{p}} \biggr). \end{aligned}$$
(2.4)
From (2.4) and Lemma 2.1, we obtain estimate (2.1). □
The first four central moments for \(K_{m}^{ ( \alpha,\beta )}\) are as follows:
$$\begin{aligned} &K_{m}^{ ( \alpha,\beta )} ( t - x;x ) = - \frac{\beta + 1}{m + \beta + 1}x + \frac{2\alpha + 1}{2 ( m + \beta + 1 )}, \\ & \begin{aligned} K_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{2};x \bigr) = {}&\frac{m - ( 2\alpha + 1 ) ( \beta + 1 )}{ ( m + \beta + 1 )^{2}}x ( 1 - x ) \\ &{}+ \frac{ ( \beta + 1 ) ( \beta - 2\alpha )}{ ( m + \beta + 1 )^{2}}x^{2} + \frac{3\alpha^{2} + 3\alpha + 1}{3 ( m + \beta + 1 )^{2}}, \end{aligned} \\ & \begin{aligned} K_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{3};x \bigr) ={}& {-} \frac{ ( 3\beta + 5 )m - ( \beta + 1 )^{3}}{ ( m + \beta + 1 )^{3}}x^{2} ( 1 - x ) \\ &{}+ \frac{ ( 12\alpha + 10 )m - 6 ( 2\alpha + 1 ) ( \beta + 1 )^{2} + 4 ( \beta + 1 )^{3}}{4 ( m + \beta + 1 )^{3}}x ( 1 - x ) \\ &{}- \frac{4 ( 3\alpha^{2} + 3\alpha + 1 ) ( \beta + 1 ) - 6 ( 2\alpha + 1 ) ( \beta + 1 )^{2} + 4 ( \beta + 1 )^{3}}{4 ( m + \beta + 1 )^{3}}x \\ &{}+ \frac{4\alpha^{3} + 6\alpha^{2} + 4\alpha + 1}{4 ( m + \beta + 1 )^{3}}, \end{aligned} \\ & K_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{4};x \bigr) \\ &\quad = \frac{3m^{2} - 2 ( 3 + 4 ( \beta + 1 ) + 3 ( \beta + 1 )^{2} )m}{ ( m + \beta + 1 )^{4}} \bigl( x ( 1 - x ) \bigr)^{2} \\ &\qquad {}- \frac{ [ 4 ( 2\alpha + 1 ) + 2 ( 6\alpha + 1 ) ( \beta + 1 ) - 6 ( \beta + 1 )^{2} ]m - 2 ( 2\alpha + 1 ) ( \beta + 1 )^{3}}{ ( m + \beta + 1 )^{4}}x^{2} ( 1 - x ) \\ &\qquad {}+ \frac{ ( 6\alpha^{2} + 10\alpha + 5 )m + 2 ( 2\alpha + 1 ) ( \beta + 1 )^{3} - 2 ( 3\alpha^{2} + 3\alpha + 1 ) ( \beta + 1 )^{2}}{ ( m + \beta + 1 )^{4}}x ( 1 - x ) \\ &\qquad {}- \frac{2 ( 2\alpha + 1 ) ( \beta + 1 )^{3} - 2 ( 3\alpha^{2} + 3\alpha + 1 ) ( \beta + 1 )^{2} + ( 4\alpha^{3} + 6\alpha^{2} + 4\alpha + 1 ) ( \beta + 1 )}{ ( m + \beta + 1 )^{4}}x \\ &\qquad {}+ \frac{5\alpha^{4} + 10\alpha^{3} + 10\alpha^{2} + 5\alpha + 1}{5 ( m + \beta + 1 )^{4}}. \end{aligned}$$

Remark 2.3

Using the results obtained by Gavrea and Ivan ([5], Theorem 14, Theorem 15, Remark 16), it is straightforward to give the following estimates:
  1. (i)
    For any \(p \ge 4\) and \(x \in ( 0,1 )\), there exists a constant \(A ( p )\) independent of m and x such that
    $$ \frac{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p + 1};x )}{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p};x )} \le \frac{A ( p )}{\sqrt{m}},\quad m \ge 5. $$
    (2.5)
     
  2. (ii)
    For any \(p \ge 1\) and \(x \in ( 0,1 )\), there exists a positive constant \(B ( p )\) independent of m and x such that
    $$ \biggl\Vert \frac{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p + 1};x )}{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p};x )} \biggr\Vert \ge \frac{B ( p )}{\sqrt{m}}, $$
    (2.6)
    where \(\Vert \cdot \Vert \) is the uniform norm on \([0,1]\).
     
  3. (iii)
    From (i) and (ii) it follows
    $$ \biggl\Vert \frac{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p + 1};x )}{K_{m}^{ ( \alpha,\beta )} ( \vert t - x \vert ^{p};x )} \biggr\Vert = O \biggl( \frac{1}{\sqrt{m}} \biggr). $$
    (2.7)
     

Remark 2.4

From Mamedov’s theorem [10] it follows that:

If \(p \in N^{ *} \) is even and \(f \in C^{p} ( [0,1] )\), for any \(x \in [0,1]\), we have that
$$ \lim_{m \to \infty} \frac{1}{K_{m}^{ ( \alpha,\beta )} ( ( t - x )^{p};x )} \Biggl( K_{m}^{ ( \alpha,\beta )} ( f;x ) - f ( x ) - \sum _{i = 1}^{p} K_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{i};x \bigr) \frac{f^{ ( i )} ( x )}{i!} \Biggr) = 0. $$
(2.8)

3 Modified Kantorovich–Stancu operators

Now, we modify the Kantorovich–Stancu operator as follows:
$$ \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) = a ( x;m )K_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) + a ( 1 - x;m )K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ). $$
(3.1)

Lemma 3.1

The moments \(\overline{K}_{m}^{ ( \alpha,\beta )} ( t^{i};x )\), \(i = 0,1,2\), are given by
$$\begin{aligned} &\overline{K}_{m}^{ ( \alpha,\beta )} ( 1;x ) = 2a_{0} ( m ) + a_{1} ( m ), \\ &\begin{aligned} \overline{K}_{m}^{ ( \alpha,\beta )} ( t;x ) ={}& \frac{ ( 2a_{0} ( m ) + a_{1} ( m ) )m}{m + \beta + 1}x \\ &{}+ \frac{4 ( \alpha + 1 )a_{0} ( m ) + ( 2\alpha + 3 )a_{1} ( m ) - 4 ( a_{0} ( m ) + a_{1} ( m ) )x}{2 ( m + \beta + 1 )}, \end{aligned} \\ &\begin{aligned} \overline{K}_{m}^{ ( \alpha,\beta )} \bigl( t^{2};x \bigr) ={}& \frac{ ( 2a_{0} ( m ) + a_{1} ( m ) )m^{2}}{ ( m + \beta + 1 )^{2}}x^{2} \\ &{}+ \frac{ [ 2 ( ( 2\alpha + 3 )a_{0} ( m ) + ( \alpha + 2 )a_{1} ( m ) )x - ( 6a_{0} ( m ) + 5a_{1} ( m ) )x^{2} ]m}{ ( m + \beta + 1 )^{2}} \\ &{}+ \frac{12 ( a_{0} ( m ) + a_{1} ( m ) )x^{2} - 6 ( a_{0} ( m ) + a_{1} ( m ) ) ( 2\alpha + 3 )x}{3 ( m + \beta + 1 )^{2}} \\ &{}+ \frac{2 ( 3\alpha^{2} + 6\alpha + 4 )a_{0} ( m ) + ( 3\alpha^{2} + 9\alpha + 7 )a_{1} ( m )}{3 ( m + \beta + 1 )^{2}}. \end{aligned} \end{aligned}$$

Lemma 3.2

The central moments of the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\), \(\overline{K}_{m}^{ ( \alpha,\beta )} ( ( t - x )^{i};x )\), \(i = 1,2,3,4\), are given by
$$\begin{aligned} &\overline{K}_{m}^{ ( \alpha,\beta )} ( t - x;x ) = - \frac{2 ( \beta + 2 )a_{0} ( m ) + ( \beta + 3 )a_{1} ( m )}{m + \beta + 1}x + \frac{4 ( \alpha + 1 )a_{0} ( m ) + ( 2\alpha + 3 )a_{1} ( m )}{2 ( m + \beta + 1 )}, \\ &\overline{K}_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{2};x \bigr) = \frac{ ( 2a_{0} ( m ) + a_{1} ( m ) )m}{ ( m + \beta + 1 )^{2}}x ( 1 - x ) + O \biggl( \frac{1}{m^{2}} \biggr), \\ &\begin{aligned} \overline{K}_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{3};x \bigr) ={}& {-} \frac{ ( ( 3\beta + 8 ) ( 2a_{0} ( m ) + a_{1} ( m ) ) + 3a_{1} ( m ) )m}{ ( m + \beta + 1 )^{3}}x^{2} ( 1 - x ) \\ &{}+ \frac{ ( ( 12\alpha + 10 ) ( 2a_{0} ( m ) + a_{1} ( m ) ) + 12 ( a_{0} ( m ) + a_{1} ( m ) ) )m}{4 ( m + \beta + 1 )^{3}}x ( 1 - x ) \\ &{}+ O \biggl( \frac{1}{m^{3}} \biggr), \end{aligned} \\ &\overline{K}_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{4};x \bigr) = \frac{3 ( 2a_{0} ( m ) + a_{1} ( m ) )m^{2}}{ ( m + \beta + 1 )^{4}} \bigl( x ( 1 - x ) \bigr)^{2} + O \biggl( \frac{1}{m^{4}} \biggr). \end{aligned}$$
We will study the uniform convergence of the sequence \(( \overline{K}_{m}^{ ( \alpha,\beta )}f )_{m \in N}\) for the case
$$ 2a_{0} ( m ) + a_{1} ( m ) = 1. $$
(3.2)
We observe that (3.2) implies \(\overline{K}_{m}^{ ( \alpha,\beta )} ( 1;x ) = 1\).

We are interested in the following cases:

Case 1:
$$ a_{0} ( m ) \ge 0\quad \mbox{and}\quad a_{0} ( m ) + a_{1} ( m ) \ge 0. $$
(3.3)
Case 2:
$$ a_{1} ( m ) < 0\quad \mbox{or}\quad a_{0} ( m ) + a_{1} ( m ) < 0. $$
(3.4)

Combining (3.2) and (3.3), we obtain \(a_{0} ( m ) \in [0,1]\) and \(a_{1} ( m ) \in [-1,1]\), which implies that the sequences \(a_{0} ( m )\) and \(a_{1} ( m )\) are bounded. The operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\) are bounded and positive.

Combining (3.2) and (3.4), we obtain that \(a_{0} ( m ) + a_{1} ( m ) > 1\) if \(a_{1} ( m ) < 0\) and \(a_{0} ( m ) > 1\) if \(a_{0} ( m ) + a_{1} ( m ) < 0\). In these cases, the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\) are not positive.

Remark that, for \(\alpha = \beta = 0\) and \(a_{0} ( m ) = \frac{3}{2}\), \(a_{1} ( m ) = - 2\), we obtain the modified operators introduced and studied in [7].

In order to prove the uniform convergence of the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\), we give the Korovkin theorem:

Theorem 3.3

([9], Theorem 10)

Let \(0 < h \in C ( [ a,b ] )\) be a function and suppose that \(( L_{n} )_{n \ge 1}\) is a sequence of positive linear operators such that \(\lim_{n \to \infty} L_{n} ( e_{i} ) = he_{i}\), \(i = 0,1,2\), uniformly on \([ a,b ]\). Then, for a given function \(f \in C ( [ a,b ] )\), we have \(\lim_{n \to \infty} L_{n} ( f ) = hf\) uniformly on \([ a,b ]\).

For the first case, we obtain the following result:

Theorem 3.4

Given two sequences \(a_{0} ( m )\) and \(a_{1} ( m )\) that satisfy conditions (3.2) and (3.3), the sequence \(( \overline{K}_{m}^{ ( \alpha,\beta )}f )_{m \in N}\) converges to f, uniformly on \([0,1]\), for any function \(f \in C ( [0,1] )\).

Proof

The operator \(\overline{K}_{m}^{ ( \alpha,\beta )}f\) is a linear convex combination of positive operators \(K_{m - 1}^{ ( \alpha,\beta + 1 )}f\) and \(K_{m - 1}^{ ( \alpha + 1,\beta + 1 )}f\). Consequently, the result follows from Theorem 3.3. □

In the second case, we have the following:

Theorem 3.5

For any function \(f \in C ( [0,1] )\) and all bounded sequences \(a_{0} ( m ), a_{1} ( m )\) that satisfy conditions (3.2) and (3.4), the sequence \(( \overline{K}_{m}^{ ( \alpha,\beta )}f )_{m \in N}\) converges to f, uniformly on \([0,1]\).

Proof

$$\begin{aligned} \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) ={}& a ( x;m )K_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) + a ( 1 - x;m )K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ) \\ ={}& \bigl( a_{1} ( m )x + a_{0} ( m ) \bigr)K_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x )\\ &{} + \bigl( - a_{1} ( m )x + a_{0} ( m ) + a_{1} ( m ) \bigr)K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ) \\ ={}& \bigl[ a_{0} ( m )K_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) + \bigl( a_{0} ( m ) - a_{1} ( m )x \bigr)K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ) \bigr] \\ &{}- \bigl[ - a_{1} ( m )xK_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) - a_{1} ( m )K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ) \bigr]. \end{aligned}$$
Taking
$$\overline{K}_{m,1}^{ ( \alpha,\beta )} ( f;x ) = a_{0} ( m )K_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) + \bigl( a_{0} ( m ) - a_{1} ( m )x \bigr)K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ) $$
and
$$\overline{K}_{m,2}^{ ( \alpha,\beta )} ( f;x ) = - a_{1} ( m )xK_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) - a_{1} ( m )K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ), $$
we have
$$ \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) = \overline{K}_{m,1}^{ ( \alpha,\beta )} ( f;x ) - \overline{K}_{m,2}^{ ( \alpha,\beta )} ( f;x ). $$
(3.5)
Using the remarks for case 2, it follows that the operators \(\overline{K}_{m,1}^{ ( \alpha,\beta )}\) and \(\overline{K}_{m,2}^{ ( \alpha,\beta )}\) are positive. According to Theorems 3.3 and 3.4, we obtain that
$$\begin{aligned} \lim_{m \to \infty} \bigl( \overline{K}_{m}^{ ( \alpha,\beta )}f \bigr) ( x ) &= \lim_{m \to \infty} \bigl( \overline{K}_{m,1}^{ ( \alpha,\beta )}f \bigr) ( x ) - \lim_{m \to \infty} \bigl( \overline{K}_{m,2}^{ ( \alpha,\beta )}f \bigr) ( x ) \\ &= ( 2l_{0} - l_{1}x )f ( x ) + l_{1} ( 1 + x )f ( x ) \\ &= ( 2l_{0} + l_{1} )f ( x ) = f ( x ), \end{aligned}$$
where \(l_{i} = \lim_{m \to \infty} a_{i} ( m )\), \(i = 0,1\). □

The following theorems are Voronovskaja-type results for the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\).

Theorem 3.6

Let \(a_{0} ( m )\), \(a_{1} ( m )\) be two convergent sequences that verify conditions (3.2) and (3.3) and \(l_{i} = \lim_{m \to \infty} a_{i} ( m )\), \(i = 0,1\). If \(f \in C^{2} ( [0,1] )\), then
$$\begin{aligned} \lim_{m \to \infty} m \bigl( \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) - f ( x ) \bigr) =& \biggl( \bigl( \alpha + 1 - ( \beta + 2 )x \bigr) + \frac{ ( 1 - 2x )l_{1}}{2} \biggr)f' ( x ) \\ &{}+ \frac{1}{2}x ( 1 - x )f'' ( x ) \end{aligned}$$
(3.6)
uniformly on \([0,1]\).

Proof

Applying Taylor’s formula to the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\), we have
$$\begin{aligned} \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) =& f ( x ) + \frac{1}{1!}\overline{K}_{m}^{ ( \alpha,\beta )} ( t - x;x )f' ( x ) + \frac{1}{2!}\overline{K}_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{2};x \bigr)f'' ( x ) \\ &{}+ \overline{K}_{m}^{ ( \alpha,\beta )} \bigl( \rho ( t;x ) ( t - x )^{2};x \bigr), \end{aligned}$$
(3.7)
where \(\rho \in C ( [0,1] )\) and \(\lim_{t \to x}\rho ( t;x ) = 0\).

It is sufficient to prove that \(\lim_{m \to \infty} m\overline{K}_{m}^{ ( \alpha,\beta )} ( \rho ( t;x ) ( t - x )^{2};x ) = 0\) uniformly on \([0,1]\).

Using the Cauchy–Schwarz theorem, we obtain that
$$\overline{K}_{m}^{ ( \alpha,\beta )} \bigl( \rho ( t;x ) ( t - x )^{2};x \bigr) \le \sqrt{\overline{K}_{m}^{ ( \alpha,\beta )} \bigl( \bigl( \rho ( t;x ) \bigr)^{2};x \bigr) \cdot \overline{K}_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{4};x \bigr)}. $$
Since \(\rho ( x,x ) = 0, \rho^{2} ( \cdot;x ) \in C ( [0,1] )\), by Theorem 3.4, we have
$$\lim_{m \to \infty} \overline{K}_{m}^{ ( \alpha,\beta )} \bigl( \bigl( \rho ( t;x ) \bigr)^{2};x \bigr) = 0, $$
and by Lemma 3.2, we get
$$\lim_{m \to \infty} m^{2}\overline{K}_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{4};x \bigr) = 3 \bigl( x ( 1 - x ) \bigr)^{2} $$
uniformly on \([0,1]\). Hence, we obtain the above limit.

Finally, Lemma 3.2 gives us (3.6). □

Theorem 3.7

Let \(a_{0} ( m )\), \(a_{1} ( m )\) be two bounded convergent sequences that verify conditions (3.2) and (3.4) and \(l_{i} = \lim_{m \to \infty} a_{i} ( m )\), \(i = 0,1\). If \(f \in C^{2} ( [0,1] )\), then
$$\lim_{m \to \infty} m \bigl( \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) - f ( x ) \bigr) = \biggl( \bigl( \alpha + 1 - ( \beta + 2 )x \bigr) + \frac{ ( 1 - 2x )l_{1}}{2} \biggr)f' ( x ) + \frac{1}{2}x ( 1 - x )f^{\prime\prime} ( x ) $$
uniformly on \([0,1]\).

Proof

From (3.5), we have
$$\overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) = \overline{K}_{m,1}^{ ( \alpha,\beta )} ( f;x ) - \overline{K}_{m,2}^{ ( \alpha,\beta )} ( f;x ), $$
where
$$\overline{K}_{m,1}^{ ( \alpha,\beta )} ( f;x ) = a_{0} ( m )K_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) + \bigl( a_{0} ( m ) - a_{1} ( m )x \bigr)K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ) $$
and
$$\overline{K}_{m,2}^{ ( \alpha,\beta )} ( f;x ) = - a_{1} ( m )xK_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) - a_{1} ( m )K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ). $$
Applying Theorem 3.6 to the operators \(\overline{K}_{m,2}^{ ( \alpha,\beta )}\) and \(\overline{K}_{m,1}^{ ( \alpha,\beta )}\), we obtain
$$\begin{aligned} \lim_{m \to \infty} m \bigl( \overline{K}_{m,1}^{ ( \alpha,\beta )} ( f;x ) - f ( x ) \bigr) =& \biggl( - ( 2l_{0} - l_{1}x ) ( \beta + 2 )x + \frac{4 ( \alpha + 1 )l_{0} - ( 2\alpha + 3 )l_{1}x}{2} \biggr)f' ( x ) \\ &{}+ \frac{1}{2} ( 2l_{0} - l_{1}x )x ( 1 - x )f'' ( x ) \end{aligned}$$
and
$$\begin{aligned} \lim_{m \to \infty} m \bigl( \overline{K}_{m,2}^{ ( \alpha,\beta )} ( f;x ) - f ( x ) \bigr) =& \biggl( l_{1} ( x + 1 ) ( \beta + 2 )x - \frac{ ( 2\alpha + 1 )x + ( 2\alpha + 3 )}{2}l_{1} \biggr)f' ( x ) \\ &{}- \frac{1}{2}l_{1} ( x + 1 )x ( 1 - x )f'' ( x ) \end{aligned}$$
uniformly on \([0,1]\).

Combining these two results, the proof is finished. □

In what follows, we will denote by \(\omega ( f; \cdot )\) the first order modulus of continuity of the function f
$$\omega ( f;\delta ) = \sup \bigl\{ \bigl\vert f \bigl( x' \bigr) - f \bigl( x^{\prime\prime} \bigr) \bigr\vert |x',x^{\prime\prime} \in I, \bigl\vert x' - x^{\prime\prime} \bigr\vert \le \delta \bigr\} ,\quad \mbox{where } I = [0,1], f:I \to R. $$

Theorem 3.8

Let \(a_{0} ( m ), a_{1} ( m )\) be two bounded sequences that verify (3.2). If \(f ( x )\) is bounded for \(x \in [0,1]\), then
$$ \bigl\Vert \overline{K}_{m}^{ ( \alpha,\beta )}f - f \bigr\Vert \le \frac{3}{2} \bigl( 3 \bigl\vert a_{1} ( m ) \bigr\vert + 1 \bigr)\omega \biggl( f;\frac{1}{\sqrt{m + \beta + 1}} \biggr), $$
(3.8)
where \(\Vert \cdot \Vert \) is the uniform norm on \([0,1]\).

Proof

By (3.1), we have that
$$\begin{aligned} \bigl\vert \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) - f ( x ) \bigr\vert \le& \bigl\vert a ( x;m ) \bigr\vert \bigl\vert K_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) - f ( x ) \bigr\vert \\ &{}+ \bigl\vert a ( 1 - x;m ) \bigr\vert \bigl\vert K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ) - f ( x ) \bigr\vert . \end{aligned}$$
(3.9)
We need an upper bound for \(a ( x;m )\) and \(a ( 1 - x;m )\). Note that this is the same upper bound for both. From (3.2), it follows that
$$\begin{aligned} \bigl\vert a ( x;m ) \bigr\vert =& \bigl\vert a_{1} ( m )x + a_{0} ( m ) \bigr\vert \le \bigl\vert a_{1} ( m ) \bigr\vert + \bigl\vert a_{0} ( m ) \bigr\vert \\ =& \bigl\vert a_{1} ( m ) \bigr\vert + \biggl\vert \frac{1 - a_{1} ( m )}{2} \biggr\vert \le \frac{3 \vert a_{1} ( m ) \vert + 1}{2} \end{aligned}$$
and (3.9) becomes
$$\begin{aligned} \bigl\vert \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) - f ( x ) \bigr\vert \le& \frac{3 \vert a_{1} ( m ) \vert + 1}{2} \bigl[ \bigl\vert K_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) - f ( x ) \bigr\vert \\ &{} + \bigl\vert K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ) - f ( x ) \bigr\vert \bigr]. \end{aligned}$$
(3.10)
By ([2], Theorem 2.6), we have
$$\bigl\vert K_{m - 1}^{ ( \alpha,\beta + 1 )} ( f;x ) - f ( x ) \bigr\vert \le 2\omega \Bigl( f;\sqrt{\delta_{m - 1,1}^{ ( \alpha,\beta + 1 )}} \Bigr) $$
and
$$\bigl\vert K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} ( f;x ) - f ( x ) \bigr\vert \le 2\omega \Bigl( f;\sqrt{\delta_{m - 1,1}^{ ( \alpha + 1,\beta + 1 )}} \Bigr), $$
where
$$\begin{aligned} &\begin{aligned} \delta_{m - 1,1}^{ ( \alpha,\beta + 1 )} ={}& \frac{ ( \beta + 2 )^{2}}{ ( m + \beta + 1 )^{2}} + \frac{ ( 2\alpha + 1 ) ( m - 1 )^{2}}{ ( m + \beta ) ( m + \beta + 1 )^{2}} + \frac{m - 1}{4 ( m + \beta + 1 )^{2}} \\ &{}+ \frac{3\alpha^{2} ( 3m + \beta - 2 ) + ( m + \beta ) ( 1 - 3m - 3\beta )}{3 ( m + \beta ) ( m + \beta + 1 )^{2}}, \end{aligned} \\ &\begin{aligned} \delta_{m - 1,1}^{ ( \alpha + 1,\beta + 1 )} ={}& \frac{ ( \beta + 2 )^{2}}{ ( m + \beta + 1 )^{2}} + \frac{ ( 2\alpha + 3 ) ( m - 1 )^{2}}{ ( m + \beta ) ( m + \beta + 1 )^{2}} + \frac{m - 1}{4 ( m + \beta + 1 )^{2}} \\ &{}+ \frac{3 ( \alpha + 1 )^{2} ( 3m + \beta - 2 ) + ( m + \beta ) ( 1 - 3m - 3\beta )}{3 ( m + \beta ) ( m + \beta + 1 )^{2}}. \end{aligned} \end{aligned}$$
So,
$$ \bigl\vert \overline{K}_{m}^{ ( \alpha,\beta )} ( f;x ) - f ( x ) \bigr\vert \le \bigl( 3 \bigl\vert a_{1} ( m ) \bigr\vert + 1 \bigr) \Bigl( \omega \Bigl( f;\sqrt{\delta_{m - 1,1}^{ ( \alpha,\beta + 1 )}} \Bigr) + \omega \Bigl( f;\sqrt{\delta_{m - 1,1}^{ ( \alpha + 1,\beta + 1 )}} \Bigr) \Bigr). $$
(3.11)
By using the properties of the first order modulus of continuity together with the above forms of \(\delta_{m - 1,1}^{ ( \alpha,\beta + 1 )}\)and \(\delta_{m - 1,1}^{ ( \alpha + 1,\beta + 1 )}\) in (3.11), we obtain (3.8). □

Assume that \(\beta = 2\alpha\), \(\overline{K}_{m}^{ ( \alpha,\beta )} ( 1;x ) = 1\) and \(\overline{K}_{m}^{ ( \alpha,\beta )} ( t;x ) = x\).

Consequently, we get
$$2a_{0} ( m ) + a_{1} ( m ) = 1\quad \mbox{and}\quad a_{0} ( m ) + a_{1} ( m ) = - \frac{2\alpha + 1}{2}, $$
which implies that
$$a_{0} ( m ) = \frac{2\alpha + 3}{2},\qquad a_{1} ( m ) = - 2 ( \alpha + 1 ) $$
and from (3.4), it follows that the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\) are not positive.

Now, we can formulate a new quantitative Voronovskaja-type result:

Theorem 3.9

For \(g \in C^{2} ( [0,1] )\), \(x \in [0,1]\) fixed, we have the following estimate:
$$ \biggl\vert \overline{K}_{m}^{ ( \alpha,\beta )} ( g;x ) - g ( x ) - \frac{1}{2}\overline{K}_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{2};x \bigr)g'' ( x ) \biggr\vert \le C \frac{1}{m}\omega \biggl( g'';\frac{1}{\sqrt{m + \beta + 1}} \biggr), $$
(3.12)
where C is a positive constant independent of m and x.

Proof

Under the above assumptions, by applying Taylor’s formula to the operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\), we have
$$ \overline{K}_{m}^{ ( \alpha,\beta )} ( g;x ) = g ( x ) + \frac{1}{2!} \overline{K}_{m}^{ ( \alpha,\beta )} \bigl( ( t - x )^{2};x \bigr)g'' ( x ) + \overline{K}_{m}^{ ( \alpha,\beta )} \bigl( r ( t;x );x \bigr), $$
(3.13)
where
$$r ( t;x ) = \int_{x}^{t} ( t - u ) \bigl[ g'' ( t ) - g'' ( u ) \bigr]\,du. $$
From the mean value theorem, it follows that there exists \(\xi \in ( \min ( x,t ),\max ( x,t ) )\) such that
$$r ( t;x ) = \bigl[ g'' ( x ) - g'' ( \xi ) \bigr] \int_{x}^{t} ( t - u )\,du = \bigl[ g'' ( x ) - g'' ( \xi ) \bigr]\frac{ ( t - x )^{2}}{2}. $$
So,
$$\begin{aligned} \bigl\vert r ( t;x ) \bigr\vert \le& \omega \bigl( g''; \vert t - x \vert \bigr)\frac{ ( t - x )^{2}}{2} \\ \le& \bigl( 1 + \sqrt{m + \beta + 1} \vert t - x \vert \bigr)\omega \biggl( g''; \frac{1}{\sqrt{m + \beta + 1}} \biggr)\frac{ ( t - x )^{2}}{2}. \end{aligned}$$
(3.14)
When \(x \in [0,1]\), an upper bound for \(a ( x;m )\) and \(a ( 1 - x;m )\) is
$$ \bigl\vert a ( x;m ) \bigr\vert = \bigl\vert a_{1} ( m )x + a_{0} ( m ) \bigr\vert = \biggl\vert - 2 ( \alpha + 1 )x + \frac{2\alpha + 3}{2} \biggr\vert \le \frac{2\alpha + 3}{2}. $$
(3.15)
From (3.15) and (3.1), we get
$$ \bigl\vert \overline{K}_{m}^{ ( \alpha,\beta )} \bigl( r ( t;x );x \bigr) \bigr\vert \le \frac{2\alpha + 3}{2} \bigl\vert K_{m - 1}^{ ( \alpha,\beta + 1 )} \bigl( r ( t;x );x \bigr) + K_{m - 1}^{ ( \alpha + 1,\beta + 1 )} \bigl( r ( t;x );x \bigr) \bigr\vert . $$
(3.16)
Using (3.14), it follows that
$$\begin{aligned} &\bigl\vert K_{m - 1}^{ ( \alpha,\beta + 1 )} \bigl( r ( t;x );x \bigr) \bigr\vert \\ &\quad \le K_{m - 1}^{ ( \alpha,\beta + 1 )} \biggl( \bigl( 1 + \sqrt{m + \beta + 1} \vert t - x \vert \bigr)\omega \biggl( g'';\frac{1}{\sqrt{m + \beta + 1}} \biggr)\frac{ ( t - x )^{2}}{2};x \biggr) \\ &\quad \le \frac{1}{2}\omega \biggl( g''; \frac{1}{\sqrt{m + \beta + 1}} \biggr) \bigl[ K_{m - 1}^{ ( \alpha,\beta + 1 )} \bigl( ( t - x )^{2};x \bigr) \\ &\qquad {}+ \sqrt{m + \beta + 1} K_{m - 1}^{ ( \alpha,\beta + 1 )} \bigl( \vert t - x \vert ( t - x )^{2};x \bigr) \bigr]. \end{aligned}$$
(3.17)
Applying Corollary 2.2, it follows that there exists a constant \(C'\) independent of m and x such that (3.17) becomes
$$ \bigl\vert K_{m - 1}^{ ( \alpha,\beta + 1 )} \bigl( r ( t;x );x \bigr) \bigr\vert \le C'\frac{1}{m - 1}\omega \biggl( g''; \frac{1}{\sqrt{m + \beta + 1}} \biggr). $$
(3.18)
Thus,
$$ \bigl\vert \overline{K}_{m}^{ ( \alpha,\beta )} \bigl( r ( t;x );x \bigr) \bigr\vert \le C\frac{1}{m}\omega \biggl( g''; \frac{1}{\sqrt{m + \beta + 1}} \biggr), $$
(3.19)
and the proof is completed. □

Corollary 3.10

For \(g \in C^{2} ( [0,1] )\), \(x \in [0,1]\) fixed, we have
$$ \lim_{m \to \infty} m \bigl( \overline{K}_{m}^{ ( \alpha,\beta )} ( g;x ) - g ( x ) \bigr) = \frac{1}{2}x ( 1 - x )g'' ( x ). $$
(3.20)

Proof

By Theorem 3.9 and Lemma 3.2(ii), we obtain (3.20). □

Corollary 3.11

For \(g \in C^{2} ( [0,1] )\), the following estimate holds:
$$ \bigl\Vert \overline{K}_{m}^{ ( \alpha,\beta )}g - g \bigr\Vert \le \frac{C}{m} \bigl\Vert g'' \bigr\Vert , $$
(3.21)
where \(\Vert \cdot \Vert \) is the uniform norm on \([0,1]\).

Proof

Since \(\omega ( g'';\delta ) \le 2 \Vert g'' \Vert \), by Lemma 3.2(ii) and Theorem 3.9, we obtain (3.21). □

We can reformulate Theorem 3.9 in terms of second order moduli of continuity.

Theorem 3.12

Assuming \(\beta = 2\alpha\), for \(a_{0} ( m ) = \frac{2\alpha + 3}{2}\), \(a_{1} ( m ) = - 2 ( \alpha + 1 )\), and \(g \in C ( [0,1] )\), we have
$$ \bigl\Vert \overline{K}_{m}^{ ( \alpha,\beta )}g - g \bigr\Vert \le C \omega_{2} \biggl( g;\frac{1}{\sqrt{m}} \biggr). $$
(3.22)

Proof

The operators \(\overline{K}_{m}^{ ( \alpha,\beta )}\)are bounded, and by (3.1), we have
$$\bigl\Vert \overline{K}_{m}^{ ( \alpha,\beta )}g \bigr\Vert \le \bigl( \bigl\vert a_{0} ( m ) \bigr\vert + \bigl\vert a_{1} ( m ) \bigr\vert \bigr) \Vert g \Vert . $$
It is well known that the second order modulus of continuity is equivalent to the K-functional
$$K_{2} \bigl( g,t^{2} \bigr) = \inf_{h \in C^{2} ( [0,1] )} \bigl\{ \Vert g - h \Vert + t^{2} \bigl\Vert h'' \bigr\Vert \bigr\} . $$
From Gonska ([6], Corollary 2.7),
$$K_{2} \bigl( g,t^{2} \bigr) \le \frac{7}{2} \omega_{2} ( g,t ),\quad t \ge 0, g \in C \bigl( [0,1] \bigr). $$
Combining the above inequalities and taking the infimum over all \(h \in C^{2} ( [0,1] )\) in the following inequality
$$\begin{aligned} \bigl\Vert \overline{K}_{m}^{ ( \alpha,\beta )}g - g \bigr\Vert &\le \bigl\Vert \overline{K}_{m}^{ ( \alpha,\beta )} ( g - h ) - ( g - h ) \bigr\Vert + \bigl\Vert \overline{K}_{m}^{ ( \alpha,\beta )}h - h \bigr\Vert \\ &\le C_{1} \Vert g - h \Vert + \frac{C_{2}}{m} \bigl\Vert g'' \bigr\Vert \le C_{3} \biggl\{ \Vert g - h \Vert + \frac{1}{m} \bigl\Vert g'' \bigr\Vert \biggr\} \end{aligned}$$
leads to the desired result. □

4 Conclusions

In this paper, we introduce and study a modified form of the Kantorovich–Stancu operators.

Declarations

Acknowledgements

The author is grateful to the PhD coordinator, Prof. Ioan Gavrea, Department of Mathematics, Technical University of Cluj-Napoca, Romania. Also, the author would like to thank the anonymous reviewers for their careful reading of the manuscript and their recommendations which improved the quality of the paper.

Funding

Not applicable.

Authors’ contributions

The author read and approved the final manuscript.

Competing interests

The author says he has no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Roumania

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