A Summation-Integral type modification of Szász-Mirakjan-Stancu operators

Vishnu Narayan Mishra\(^\ast \), Rajiv B. Gandhi\(^{\S }\) Ram N. Mohapatra\(^\bullet \)

January 12, 2016.

\(^\ast , ^{\S }\)Applied Mathematics & Humanities Department, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev, Dumas Road, Surat - 395 007, (Gujarat), India

\(^\ast \)L. 1627 Awadh Puri Colony Beniganj, Phase-III, Opposite - Industrial Training Institute (ITI), Ayodhya Main Road, Faizabad, Uttar Pradesh 224 001, India, e-mail: vishnunarayanmishra@gmail.com, vishnu_narayanmishra@yahoo.co.in

\(^\S \)Department of Mathematics, BVM Engineering College, Vallabh Vidyanagar - 388 120, (Gujarat), India, e-mail: rajiv55in@yahoo.com. \(^\bullet \)Department of Mathematics, University of Central Florida, Orlando, FL. 32816, USA, e-mail: Ram.mohapatra@ucf.edu

In this paper, we introduce a summation-integral type modification of Szasz-Mirakjan-Stancu operators. Calculation of moments, density theorem, a direct result and a Voronovskaja-type result are obtained for the operators.

MSC. 41A10, 41A25, 41A36, 40A30

Keywords. Szász-Mirakjan operators, the Korovkin-type approximation result, K-functional, modulus of smoothness, Voronovskaja-type result.

1 Introduction

In 1941, G.M. Mirakjan [ 17 ] defined the operators \(SM_n: C_2[0,\infty ) \rightarrow C[0,\infty )\) for any \(x \in [0, \infty )\) and for any \(n \in \mathbb {N}\) given by,

\begin{equation} \label{E1} SM_n(f;x)=\sum _{k=0}^{\infty }s_{n,k}(x) f\left(\tfrac {k}{n}\right), \end{equation}
1

where

\begin{equation} \label{E2} s_{n,k}(x)=e^{-nx}\tfrac {(nx)^k}{k!}, \qquad 0 \leq x < \infty . \end{equation}
2

and

\begin{equation*} C_2[0,\infty ) = \left\lbrace f \in C[0,\infty ) : \lim _{x \rightarrow \infty } \tfrac {f(x)}{1+x^2} \, \text{exists and is finite} \right\rbrace . \end{equation*}

The operators \(\left(SM_n \right)_{n \in \mathbb {N}}\) are named Szász-Mirakjan operators, where \(s_{n,k}\)’s are Szász basis functions. They were extensively studied in 1950 by O. Szász [ 19 ] . A modification of operators (1) was introduced by P.L. Butzer [ 7 ] , in order to obtained an approximation process for spaces of integrable functions, on unbounded intervals, which are now known as Szász-Mirakjan-Kantorovich operators.

Durrmeyer [ 9 ] defined the summation-integral type approximation process, using the Bernstein polynomials, as

\begin{equation} \label{Dur} D_n(f;x)=(n+1)\sum _{k=0}^n b_{n,k}(x)\left(\int _{0}^1 b_{n,k}(t)f(t)\, dt \right), \end{equation}
3

where Bernstein polynomial are given by

\begin{equation*} B_n(f;x)=\sum _{k=0}^n b_{n,k}(x)f\left(\tfrac {k}{n} \right), \end{equation*}

and

\begin{equation*} b_{n,k}(x) = \tbinom {n}{k}x^k(1-x)^{n-k}, \end{equation*}

\(0 \leq x \leq 1\), \(k= 0,1, \ldots , n\) and \(n \in \mathbb {N}\).

Derriennic [ 8 ] studied the operators given by (3) extensively. Motivated by Derriennic, Sahai and Prasad [ 18 ] studied many properties of the modified Lupaş operators of the type

\begin{equation} \label{Sah} M_n(f;x)=(n-1)\sum _{k=0}^{\infty }p_{n,k}(x)\left(\int _0^{\infty } p_{n,k}(t)f(t)\, dt \right), \end{equation}
4

where

\begin{equation*} p_{n,k}(x)=\tbinom {n+k-1}{k}\tfrac {x^k}{(1+x)^{n+k}}, \end{equation*}

\(0 \leq x {\lt} \infty \), \(k= 0,1,2, \ldots \) and \(n \in \mathbb {N}\).

Mazhar and Totik [ 16 ] introduced two Durrmeyer type modifications of Szász-Mirakjan operators (1) as

\begin{equation*} \bar{S}_n(f;x)=f(0)s_{n,0}(x)+ n\sum _{k=1}^{\infty }s_{n,k}(x)\int _{0}^{\infty }s_{n,k-1}(t)f(t)\, dt \end{equation*}

and

\begin{equation} \label{Maz1} S_n(f;x)= n\sum _{k=0}^{\infty }s_{n,k}(x)\int _{0}^{\infty }s_{n,k}(t)f(t)\, dt, \end{equation}
5

where \(s_{n,k}\)’s are as given by (2).

Various properties, like global approximation in weight spaces, uniform approximation, simultaneous approximation, weighted approximations, of these operators, their generalizations and modifications are studied over the years. We can mention some important studies of this type (see [ 1 ] [ 3 ] , [ 5 ] [ 15 ] , [ 21 ] [ 23 ] , [ 25 ] [ 32 ] ).

In 2015, Mishra et al. [ 24 ] introduced Szász-Mirakjan-Durrmeyer-type generalization of (5) given by

\begin{equation} \label{RBG} S_n^{*}(f;x) = b_n\sum _{k=0}^{\infty } s_{b_n,k}(x)\int _0^{\infty }s_{b_n,k}(t)f\left(t\right)\, dt, \end{equation}
6

where

\begin{equation} \label{Sza} s_{b_n,k}(x)=e^{-b_nx}\tfrac {(b_nx)^k}{k!}, \quad k = 0,1,2, \ldots ; n \in \mathbb {N}, \end{equation}
7

\((b_n)\) is an increasing sequence of positive real numbers, \( b_n \rightarrow \infty \) as \(n \rightarrow \infty \), \(b_1 \geq 1\) and studied the simultaneous approximation properties of the operators (6).

In this article, we introduce Stancu [ 20 ] type summation integral type of modification for \(0 \leq \alpha \leq \beta \), for any \(x \in [0, \infty )\) and for any \(n \in \mathbb {N}\) of (\(\ref{E1}\)) given by

\begin{equation} \label{E4} S_n^{*,\alpha , \beta }(f;x)= b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) f\big(\tfrac {b_nt + \alpha }{b_n + \beta }\big)\, dt, \end{equation}
8

where \(s_{b_n,k}\)’s are as given in (\(\ref{Sza}\)), \((b_n)\) is an increasing sequence of positive real numbers with \(b_n \rightarrow \infty \) as \(n \rightarrow \infty \) and \(b_1 + \beta \geq 1\). The operators \(\big(S_n^{*,\alpha , \beta }\big)_{n \in \mathbb {N}}\), given by (8), are positive linear operators. Here we studied the direct result related properties of the operators (8).

2 Estimation of moments

Let us denote by \(\mu _{n,m}^{*,\alpha ,\beta }\), the \(m^{th}\) moments of the operators given in (8), defined as

\begin{equation} \label{E5} \mu _{n,m}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x)^m;x),\qquad m = 0,1,2,\ldots . \end{equation}
9

Lemma 1

For \(m=1,2, \ldots \), the following relation holds:

\begin{eqnarray} \label{E11} S_n^{*,\alpha , \beta }(t^m;x)=\sum _{j=0}^{m} \tbinom {m}{j}\tfrac {b_n^j \alpha ^{m-j}}{(b_n + \beta )^m}S_n^{*}(t^j;x), \end{eqnarray}

where \(S_n^{*}(f;x)\) and \(S_n^{*,\alpha , \beta }(f;x)\) are as given by (\(\ref{RBG}\)) and (\(\ref{E4}\)), respectively.

Proof â–¼
Using (\(\ref{RBG}\)) and (8),

\begin{eqnarray*} S_n^{*,\alpha , \beta }(t^m;x)& =& b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left(\tfrac {b_nt + \alpha }{b_n + \beta }\right)^m\, dt\\ & =& b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left(\sum _{j=0}^{m} \tbinom {m}{j} \tfrac {b_n^j \alpha ^{m-j}}{(b_n + \beta )^m} t^j \right)\, dt \\ & =& \sum _{j=0}^{m} \tbinom {m}{j} \tfrac {b_n^j \alpha ^{m-j}}{(b_n + \beta )^m}\left( b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) t^j \, dt \right) \\ & =& \sum _{j=0}^{m} \tbinom {m}{j} \tfrac {b_n^j \alpha ^{m-j}}{(b_n + \beta )^m}S_n^{*}(t^j;x). \end{eqnarray*}

Lemma 2

For \(m=1,2, \ldots \), the following holds:

\begin{eqnarray*} S_n^{*,\alpha , \beta }((t-x)^m;x)= \sum _{j=0}^{m} \tbinom {m}{j} (-x)^{m-j}S_n^{*,\alpha , \beta }(t^j;x), \end{eqnarray*}

where \(S_n^{*,\alpha , \beta }(f;x)\) is as given in (\(\ref{E4}\)).

Proof â–¼
Using (\(\ref{E4}\)),

\begin{align*} & S_n^{*,\alpha , \beta }((t-x)^m;x)=\\ & = b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left(\tfrac {b_nt + \alpha }{b_n + \beta } - x\right)^m\, dt \\ & = b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left\lbrace \sum _{j=0}^{m} \tbinom {m}{j} (-x)^{m-j} \left(\tfrac {b_nt+\alpha }{b_n+\beta } \right)^j \right\rbrace \, dt \\ & =\sum _{j=0}^{m} \tbinom {m}{j} (-x)^{m-j}\left\lbrace b_n \sum _{k=0}^{\infty }s_{b_n,k}(x)\int _{0}^{\infty }s_{b_n,k}(t) \left(\tfrac {b_nt+\alpha }{b_n+\beta } \right)^j \, dt \right\rbrace \\ & =\sum _{j=0}^{m} \tbinom {m}{j} (-x)^{m-j}S_n^{*,\alpha , \beta }(t^j;x). \end{align*}

Lemma 3

For \(e_i(t)=t^i, \quad i = 0,1,2,3,4\), the following holds:

  1. \(S_n^{*}(e_0;x)=1,\)

  2. \(S_n^{*}(e_1;x)=\tfrac {1}{b_n}(b_nx+1),\)

  3. \(S_n^{*}(e_2;x)=\tfrac {1}{b_n^2}(b_n^2x^2+4b_nx+2),\)

  4. \(S_n^{*}(e_3;x)=\tfrac {1}{b_n^3}(b_n^3x^3+9b_n^2x^2+18b_nx+6),\)

  5. \(S_n^{*}(e_4;x)=\tfrac {1}{b_n^4}(b_n^4x^4 + 16b_n^3x^3+72b_n^2x^2+96b_nx+24).\)

Proof â–¼
By (\(\ref{RBG}\)),

\begin{eqnarray*} S_n^{*}(e_2;x)& =& \sum _{k=0}^{\infty }s_{b_n,k}(x) b_n \int _{0}^{\infty }s_{b_n,k}(t) t^2\, dt \\ & =& \sum _{k=0}^{\infty }s_{b_n,k}(x) b_n \int _{0}^{\infty }e^{-b_nt} \tfrac {(b_nt)^k}{k!} t^2\, dt \\ & =& \sum _{k=0}^{\infty }s_{b_n,k}(x) \tfrac {(k+1)(k+2)}{b_n^2}\\ & =& \tfrac {1}{b_n^2}\sum _{k=0}^{\infty }e^{-b_nx} \tfrac {(b_nx)^k}{k!} [k(k-1)+4k+2] \\ & =& \tfrac {1}{b_n^2}(b_n^2x^2+4b_nx+2). \end{eqnarray*}

This proves (c).

Other relations follow on the same line.

Lemma 4

For \(e_i(t)=t^i, \quad i = 0,1,2,3,4\), the following holds:

  1. \(S_n^{*,\alpha , \beta }(e_0;x)=1,\)

  2. \(S_n^{*,\alpha , \beta }(e_1;x)=\tfrac {1}{b_n+\beta }(b_nx+\alpha +1),\)

  3. \(S_n^{*,\alpha , \beta }(e_2;x)=\tfrac {1}{(b_n+\beta )^2}[b_n^2x^2+2b_nx(\alpha +2)+(\alpha ^2 + 2\alpha + 2)],\)

  4. \(S_n^{*,\alpha , \beta }(e_3;x)=\tfrac {1}{(b_n+\beta )^3}\Big[b_n^3x^3+3b_n^2x^2(\alpha +3)+3b_nx(\alpha ^2+4\alpha +6)+(\alpha ^3+3\alpha ^2 + 6\alpha + 6)\Big],\)

  5. \(S_n^{*,\alpha , \beta }(e_4;x)=\tfrac {1}{(b_n+\beta )^4}\Big[b_n^4x^4+ 4b_n^3x^3(\alpha +4)+6b_n^2x^2(\alpha ^2+6\alpha +12)+4b_nx(\alpha ^3+6\alpha ^2+18\alpha +24)+(\alpha ^4+4\alpha ^3+12\alpha ^2 + 24\alpha + 24)\Big].\)

Proof â–¼
Using (10),

\begin{eqnarray*} S_n^{*,\alpha , \beta }(e_2;x)& =& \sum _{j=0}^{2} \tbinom {2}{j} \tfrac {b_n^j \alpha ^{2-j}}{(b_n + \beta )^2}S_n^{*}(t^j;x)\\ & =& \tfrac {1}{(b_n+\beta )^2}[\alpha ^2 S_n^{*}(1;x)+2b_n\alpha S_n^{*}(t;x)+b_n^2 S_n^{*}(t^2;x)]\\ & =& \tfrac {1}{(b_n+\beta )^2}[b_n^2x^2+2b_nx(\alpha +2)+(\alpha ^2 + 2\alpha + 2)]. \end{eqnarray*}

This proves (c).

Other relations follow on the same line. Consider the Banach lattice

\begin{equation*} C_\gamma [0,\infty ) = \Big\{ f \in C[0,\infty ): \left|f(t) \right| \leq M (1+t)^{\gamma }\Big\} \end{equation*}

for some \(M{\gt}0, \gamma {\gt}0\).

Theorem 5

\(\lim _{n \rightarrow \infty } S_n^{*,\alpha , \beta }(f;x)= f(x)\) uniformly for \(x \in [0,a]\), provided \(f \in C_\gamma [0,\infty )\), \(\gamma \geq 2\) and \(a {\gt} 0\).

Proof â–¼
For fix \(a {\gt}0\), consider the lattice homomorphism \(T_a: C[0, \infty ) \rightarrow C[0,a]\) defined by \(T_a(f):= \left.f\right|_{[0,a]}\) for every \(f \in C[0, \infty )\), where \(\left.f\right|_{[0,a]}\) denotes the restriction of the domain of \(f\) to the interval \([0,a]\). In this case, we see that, for each \(i = 0,1,2\) and by (a)–(c) of Lemma 4,

\begin{equation} \label{E13} \lim _{n \rightarrow \infty } T_a \left(S_n^{\alpha , \beta }(e_i;x)\right)= T_a( e_i(x)), \; \text{uniformly on} \; [0,a]. \end{equation}
11

Thus, by using (11) and with the universal Korovkin-type property with respect to positive linear operators (see Theorem 4.1.4 (vi) of [ 4 ] , p.199) we have the result.

Lemma 6

For the moments defined in (\(\ref{E5}\)), the following holds:

  1. \(\mu _{n,1}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x);x)=\tfrac {1}{b_n+\beta }(\alpha +1-\beta x) ,\)

  2. \(\mu _{n,2}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x)^2;x)=\tfrac {1}{(b_n+\beta )^2}\Big(\beta ^2x^2+2x(b_n-\alpha \beta -\beta )+(\alpha ^2 + 2\alpha + 2)\Big),\)

  3. \(\mu _{n,3}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x)^3;x)=\tfrac {1}{(b_n+\beta )^3}\Big[-\beta ^3x^3+3\beta x^2 \{ (2b_n+\beta )(\alpha +1)-2(\alpha +2)\} +3x \{ 2b_n(\alpha +2)-\beta (\alpha ^2+2\alpha +2)\} +(\alpha ^3+3\alpha ^2 + 6\alpha + 6)\Big],\)

  4. \(\mu _{n,4}^{*,\alpha ,\beta }(x)= S_n^{*,\alpha , \beta }((t-x)^4;x)=\tfrac {1}{(b_n+\beta )^4}\Big[\beta ^4 x^4 + 4\beta ^2x^3 \{ 3b_n - \beta (\alpha +1)\} +6x^2 \{ 2b_n^2 -4b_n \beta (\alpha +3) + \beta ^2 (\alpha ^2 + 2\alpha + 2)\} + 4x \{ 3b_n(\alpha ^2+4\alpha +6)-\beta (\alpha ^3 + 3 \alpha ^2 + 6 \alpha + 6)\} + (\alpha ^4 +4\alpha ^3 + 12 \alpha ^2 + 24 \alpha + 24)\Big].\)

Proof â–¼

The results follow from linearity of the operators \(S_n^{*,\alpha , \beta }\) and lemma (\(\ref{L4}\)).

3 Direct result

Let us consider the space \(C_B[0,\infty )\) of all continuous and bounded functions on \([0, \infty )\) and for \(f \in C_B[0,\infty )\), consider the supremum norm \(\| f\| =\sup \{ |f(x)|: x \in [0,\infty )\} \). Also, consider the \(K\)-functional

\begin{equation} \label{E7} K_2(f;\delta )=\inf _{g \in W^2}\left\lbrace \| f-g \| +\delta \| g'' \| \right\rbrace , \end{equation}
12

where \(\delta {\gt}0\) and \(W^2=\left\lbrace g \in C_B[0,\infty ): g', g'' \in C_B[0, \infty ) \right\rbrace \). For a constant \(C {\gt}0\), the following relationship exists:

\begin{align} \label{E8} K_2(f;\delta )\leq C \omega _2(f, \sqrt{\delta }), \end{align}

where

\begin{align} \label{E9} \omega _2(f, \sqrt{\delta })=\sup _{0{\lt}h{\lt}\sqrt{\delta }} \; \sup _{x \in [0, \infty )}|f(x+2h)-2f(x+h)+f(x)| \end{align}

is the second order modulus of smoothness of \(f \in C_B[0,\infty )\); and for \(f \in C_B[0,\infty )\), let the modulus of continuity be given by

\begin{align} \label{E10} \omega _1(f, \sqrt{\delta })=\sup _{0{\lt}h{\lt}\sqrt{\delta }} \; \sup _{x \in [0, \infty )}|f(x+h)-f(x)|. \end{align}

Theorem 7

For \(f \in C_B[0, \infty )\), we have

\begin{equation*} |S_n^{*,\alpha , \beta }(f;x) - f(x)| \leq \omega _1 \left(f, \tfrac {\left|\alpha +1-\beta x\right|}{b_n+\beta } \right)+ C\omega _2 \left(f,\sqrt{\mu _{n,2}^{*,\alpha ,\beta }(x) + \big(\tfrac {\alpha +1-\beta x}{b_n+\beta }\big)^2} \right), \end{equation*}

where \(C\) is a positive constant.

Proof â–¼
Let the auxiliary operator denoted by \(\bar{S}_n^{*,\alpha , \beta }\) be defined as

\begin{equation*} \bar{S}_n^{*,\alpha , \beta }(f;x) = S_n^{*,\alpha , \beta }(f;x)-f\left(\tfrac {b_nx+\alpha +1}{b_n+\beta } \right)+f(x) \end{equation*}

for every \(x \in [0, \infty )\). It is a linear operator which preserves the linear functions as \(\bar{S}_n^{*,\alpha , \beta }(1;x)=1\) and \(\bar{S}_n^{*,\alpha , \beta }(t;x)=x\). This gives us \(\bar{S}_n^{*,\alpha , \beta }(t-x;x)=0\).

For \(g \in W^2\), \(x \in [0, \infty )\) and by Taylor’s expansion, we have

\begin{equation*} g\left(\tfrac {b_nt+\alpha }{b_n+\beta }\right) = g(x) + \left(\tfrac {b_nt+\alpha }{b_n+\beta }-x\right)g’(x)+\int _x^{\frac{b_nt+\alpha }{b_n+\beta }}\left(\tfrac {b_nt+\alpha }{b_n+\beta }-u\right) g”(u)du. \end{equation*}

Operating \(\bar{S}_n^{*,\alpha , \beta }\) on both the sides,

\begin{eqnarray*} \left|\bar{S}_n^{*,\alpha , \beta }(g;x)-g(x)\right| & =& \left| \bar{S}_n^{*,\alpha , \beta }\left( \int _x^t(t-u)g”(u)du;x\right)\right| \\ & \leq & \left| S_n^{*,\alpha , \beta }\left( \int _x^t(t-u)g”(u)du;x\right)\right| \\ & & + \left|\int _x^{\tfrac {b_nx+\alpha +1}{b_n+\beta }} \left(\tfrac {b_nx+\alpha +1}{b_n+\beta }-u \right) g”(u)du\right| \\ & \leq & \Vert g” \Vert S_n^{*,\alpha , \beta }((t-x)^2;x)+ \Vert g” \Vert \left(\tfrac {b_nx+\alpha +1}{b_n+\beta } -x \right)^2 \\ & = & \Vert g” \Vert \left[ \mu _{n,2}^{*,\alpha , \beta }(x)+ \left(\tfrac {- \beta x+\alpha +1}{b_n+\beta }\right)^2 \right]. \end{eqnarray*}

Also, we have \(\left|S_n^{*,\alpha , \beta }(f;x)\right| \leq \Vert f \Vert \). Using these, we get

\begin{eqnarray*} \left|S_n^{*,\alpha , \beta }(f;x)- f(x)\right|& \leq & \left|\bar{S}_n^{*,\alpha , \beta }(f-g;x)- (f-g)(x)\right| \\ & & + \left|\bar{S}_n^{*,\alpha , \beta }(g;x)- g(x)\right|+ \left|f\left( \tfrac {b_nx+\alpha +1}{b_n+\beta }\right)- f(x)\right| \\ & \leq & 2 \Vert f-g \Vert + \Vert g” \Vert \left[ \mu _{n,2}^{*,\alpha , \beta }(x)+ \left(\tfrac {- \beta x+\alpha +1}{b_n+\beta }\right)^2 \right] \\ & & + \omega _1 \left(f, \tfrac {\left|-\beta x+\alpha +1\right|}{b_n+\beta } \right). \end{eqnarray*}

Taking infimum on the right hand side for all \(g \in W^2\), we get

\begin{eqnarray*} \left|S_n^{*,\alpha , \beta }(f;x)- f(x)\right| & \leq & 2K_2\left(f,\tfrac {1}{2} \left[ \mu _{n,2}^{*,\alpha , \beta }(x)+ \left(\tfrac {- \beta x+\alpha +1}{b_n+\beta }\right)^2 \right]\right) \\ & & + \omega _1 \left(f, \tfrac {\left|-\beta x+\alpha +1\right|}{b_n+\beta } \right). \end{eqnarray*}

Using (13) and \(\omega _2(f, \lambda \delta ) \leq (\lambda + 1)^2 \omega _2(f, \delta )\) for \(\lambda {\gt}0\), we get

\begin{eqnarray*} \left|S_n^{*,\alpha , \beta }(f;x)- f(x)\right| & \leq & C \omega _2\left(f, \sqrt{\mu _{n,2}^{*,\alpha , \beta }(x)+ \left(\tfrac {- \beta x+\alpha +1}{b_n+\beta }\right)^2} \right) \\ & & + \omega _1 \left(f, \tfrac {\left|-\beta x+\alpha +1\right|}{b_n+\beta } \right). \end{eqnarray*}

4 A Voronovskaja-type result

In this section we prove a Voronovskaja-type theorem for the operators \(S_n^{*,\alpha , \beta }\) given in (8).

Lemma 8

\(\lim _{n \rightarrow \infty } (b_n+\beta )^2 \mu _{n,4}^{*,\alpha , \beta }(x) = 12 x^2\) uniformly with respect to \(x \in [0, a],\, a {\gt} 0\).

Proof â–¼

By lemma (6)(d), we may write that

\begin{eqnarray*} (b_n+\beta )^2 \mu _{n,4}^{*,\alpha ,\beta }(x)& =& \tfrac {1}{(b_n+\beta )^2}\Big[\beta ^4 x^4 + 4\beta ^2x^3 \{ 3b_n - \beta (\alpha +1)\} \\ & & +6x^2 \{ 2b_n^2 -4b_n \beta (\alpha +3) + \beta ^2 (\alpha ^2 + 2\alpha + 2)\} \\ & & + 4x \{ 3b_n(\alpha ^2+4\alpha +6)-\beta (\alpha ^3 + 3 \alpha ^2 + 6 \alpha + 6)\} \\ & & + (\alpha ^4 +4\alpha ^3 + 12 \alpha ^2 + 24 \alpha + 24)\Big] \end{eqnarray*}

Taking limits on both sides, as \(n \rightarrow \infty \), the Lemma is proved.

Theorem 9

For every \(f \in C_\gamma [0,\infty )\) such that \(f', f'' \in C_\gamma [0,\infty ), \gamma \geq 4\), we have

\begin{equation*} \lim _{n \rightarrow \infty } (b_n+\beta ) \left[S_n^{*,\alpha , \beta }(f;x)-f(x) - \tfrac {-\beta x + \alpha + 1}{b_n } f’(x) \right]= xf”(x) \end{equation*}

uniformly with respect to \(x \in \left[\tfrac {\alpha }{b_n + \beta }, a\right] \, \left(a {\gt} \tfrac {\alpha }{b_n + \beta }\right)\).

Proof â–¼
Let \(f,f', f'' \in C_\gamma [0,\infty )\) and \(x \geq \tfrac {\alpha }{b_n + \beta }\). Define, for \(t \in [0, \infty )\) with \(\tfrac {b_nt+ \alpha }{b_n + \beta } \neq x\),

\begin{align*} & \Psi \left(t,x\right) = \\ & = \tfrac {1}{\left(\frac{b_nt+\alpha }{b_n+\beta }-x\right)^2} \left[f\left(\tfrac {b_nt+\alpha }{b_n+\beta }\right)\! -\! f(x)-\left(\tfrac {b_nt+\alpha }{b_n+\beta }\! -\! x\right) f’(x) \! -\! \tfrac {1}{2}\left(\tfrac {b_nt+\alpha }{b_n+\beta }\! -\! x\right)^2 f”(x)\right] , \end{align*}

and \(\Psi \Big(\tfrac {\left(b_n + \beta \right)x - \alpha }{b_n},x\Big)=0\). The function \(\Psi (\cdot ,x) \in C_\gamma [0,\infty )\). Also, for \(n \rightarrow \infty \), \(\Psi \Big(\tfrac {\left(b_n + \beta \right)x - \alpha }{b_n},x\Big)=\Psi \left( x,x\right)\), so \(\Psi (x,x)=0\), as \(n \rightarrow \infty \). By Taylor’s theorem we get

\begin{align*} & f\left(\tfrac {b_nt+\alpha }{b_n+\beta }\right)= \\ & = f(x)+\left(\tfrac {b_nt+\alpha }{b_n+\beta }-x\right)f’(x)+\tfrac {1}{2}\left(\tfrac {b_nt+\alpha }{b_n+\beta }-x\right)^2 f”(x) +\left(\tfrac {b_nt+\alpha }{b_n+\beta }-x\right)^2\Psi (t,x). \end{align*}

Now from Lemma 6(a)–(b)

\begin{align} \label{E12} (b_n\! +\! \beta ) \left[S_n^{*,\alpha , \beta }(f;x)\! -\! f(x)\right] = & (b_n\! +\! \beta )f’(x)\mu _{n,1}^{*,\alpha , \beta }(x) \! +\! \tfrac {1}{2} (b_n\! +\! \beta )f”(x)\mu _{n,2}^{*,\alpha , \beta }(x) \\ & + (b_n+\beta ) S_{n}^{*,\alpha , \beta }((t-x)^2 \Psi (t,x)). \nonumber \end{align}

If we apply the Cauchy-Schwarz inequality to the third term on the right hand side of (16), then

\begin{equation*} (b_n+\beta ) S_{n}^{*,\alpha , \beta }((t-x)^2 \Psi (t,x);x) \leq \left( (b_n+\beta )^2 \mu _{n,4}^{*,\alpha , \beta }(x)\right)^{\frac{1}{2}} ( S_{n}^{*,\alpha , \beta }(\Psi ^2(t,x);x))^{\frac{1}{2}} \end{equation*}

Now \(\Psi ^2(\cdot ,x) \in C_{2\gamma }[0,\infty )\), using Theorem 5, we have \(S_{n}^{*,\alpha , \beta }(\Psi ^2(t,x);x) \rightarrow \Psi ^2(x,x)=0\), as \(n \rightarrow \infty \) and using Lemma 8, this third term on the right tends to zero for \(x \in \left[\tfrac {\alpha }{b_n + \beta }, a\right]\) and we get

\begin{equation*} \lim _{n \rightarrow \infty } (b_n+\beta ) \left[S_n^{*,\alpha , \beta }(f;x)-f(x)\right]= (1+\alpha - \beta x)f’(x) + xf”(x). \end{equation*}

for \(x \in \left[\tfrac {\alpha }{b_n + \beta }, a\right] \, \left(a {\gt} \tfrac {\alpha }{b_n + \beta }\right)\).

Acknowledgement

The authors are extremely grateful to the learned referee(s) and the editor for their careful reading of the manuscript, valuable suggestions and constructive comments, which has greatly contributed to improving the quality of this research article.

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