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\title[Approx. by A Gen. of Sz\'{a}sz-Mirakjan type Operators]{Approximation by A Generalization of  Sz\'{a}sz-Mirakjan type Operators}
\pagestyle{myheadings}\markboth{ M. A. Siddiqui, Nandita Gupta }{Approximation by A Generalization of  Sz\'{a}sz-Mirakjan type Operators}
 \author{ M. A. Siddiqui\textsuperscript{$1$}, Nandita Gupta\textsuperscript{$2,\ast$}  }

     \begin{document}
      \maketitle
      \thispagestyle{empty}
      \begin{center}
      \scriptsize{\textsuperscript{$1,2$} Department Of Mathematics,
Govt. V. Y. T. P. G. Autonomous College, Durg,}
\end{center}\begin{center}\scriptsize{
     Chhattisgarh, \textbf{India.}}\\
\scriptsize{\textsuperscript{$1$}
     Email:dr\_m\_a\_siddiqui@yahoo.co.in\\
     \textsuperscript{$2$}
     Email:nandita\_dec@yahoo.com\\
       \textsuperscript{$\ast$} Corresponding author }
     \end{center}
     \begin{quotation}
      \vspace{.5 in}
    \hspace{.5 cm} \scriptsize{ \textbf{Abstract}.
    In the present paper we propose a new generalization of
     Sz\'{a}sz-Mirakjan-type operators. We discuss their weighted convergence and rate of convergence via weighted modulus
of continuity. We also give an asymptotic estimate through
Voronovskaja type result for these operators.\\\\
 \noindent
\textit{\textbf{2010 AMS  Subject Classification}}. Primary 41A36.\\

\noindent \textit{\textbf{Key Words and Phrases}}. Linear positive operators, Sz\'{a}sz-Mirakjan operators, Rate of convergence, Weighted Korovkin-type
theorem, Weigh-ted Modulus of continuity.}
     \end{quotation}
 \section{Introduction}
 In \cite{les} Rempulska et al. introduced the following operators of Sz\'{a}sz-Mirakjan type
\begin{equation}\label{orgcos}
  L_n(f;x)= \sum_{k=0}^ \infty p_{n,k}(x) f\Big(\frac{2k}{n}\Big),
\end{equation}
with
\begin{equation}
p_{n,k}(x)=\frac{1}{\cosh{nx}} \frac{(nx)^{2k}}{(2k)!}, ~  ~ k \in \mathbb{N}_0 = \mathbb{N} \cup \{0\},
\end{equation}%and
where  $f \in C_B$  and \(C_B\) is the space of real-valued
functions uniformly continuous
 and bounded on \( \R^+ =[0, ~ ~\infty) \) and   the norm in
 \(C_B\) is given as \[ \|f\| = \sup_{x \in \R^+}|f(x)|.
 \]
\indent In \cite{lrms2,lrms3} a Voronovskaja-type theorem was proved for these operators.

In 2014, Aral et al. \cite{ali2} introduced a very interesting generalization of the Sz\'{a}sz-Mirakjan operators \cite{osz} using a function $\rho$ as
\begin{align}
S_n^{\rho}(f;x)=&e^{-n\rho(x)}\sum_{k=0}^{\infty}\left(f\circ\rho^{-1}\right)\left(\frac{k}{n}\right)\frac{(n\rho(x))^k}{k!}\\
=&(S_n(f\circ\rho^{-1})\circ\rho)(x)\nonumber\\
=&e^{-n\rho(x)}\sum_{k=0}^{\infty}f\left(\rho^{-1}\left(\frac{k}{n}\right)\right)\frac{(n\rho(x))^k}{k!},\nonumber
 \end{align}
 where the function $\rho$ satisfies following properties:\\
 $(\rho_1)$ $\rho$ is continuously differentiable on $\R^+$,\\
 $(\rho_2)$ $\rho(0)=0$, $\displaystyle \inf_{x \in \R^+}\rho'(x)\geq 1.$

We propose a similar generalization of the operators \eqref{orgcos} as follows
\begin{equation}\label{cosnew}
  L_n^{\rho}(f;x)= \frac{1}{\cosh{(n\rho(x))}}\sum_{k=0}^ \infty (f\circ\rho^{-1}) \left(\frac{2k}{n}\right)\frac{(n\rho(x))^{2k}}{(2k)!},
\end{equation}
%and
 %\begin{equation}\label{sinnew}
  %A_n^{\rho}(f;x)= \frac{f(0)}{(1+\sinh{n\rho(x)})}+\frac{1}{(1+\sinh{n\rho(x)})}\sum_{k=0}^ \infty (fo\rho^{-1}) \left(\frac{2k+1}{n}\right)\frac{(n\rho(x))^{2k+1}}{(2k+1)!}
%\end{equation}
where $x \in \R^+$, $n \in \N$, $k \in \N_0 = \N \cup \{0\} $ and function $\rho$ satisfies conditions $(\rho_1)$ and $(\rho_2)$.

We see that these new operators are positive linear operators. For $\rho(x) =x$, these operators \eqref{cosnew} reduce to the operators \eqref{orgcos}.
Also from conditions  $(\rho_1)$ and $(\rho_2)$ we
can draw out that\\
(i) $\displaystyle \lim_{x \in \R^+}\rho(x)=\infty$,\\
(ii) $|t-x| \leq |\rho(t)-\rho(x)|$ for all $x$, $t\in \R^+.$

In this paper we study some approximation properties of these new operators. Firstly we prove a theorem for the weighted convergence of $L_n^{\rho}f$ to
$f$ with the help of a weighted Korovkin-type theorem \cite{adg1}, \cite{adg2}. Then we determine an estimate of the rate of the weighted convergence using
weighted modulus of continuity defined in \cite{holhos}. At the end we prove a Voronovskaja type result for these new operators.
\section{Weighted Convergence of $L_n^{\rho}(f;x)$ }

From the definition of the operators $L_n^{\rho}$ one can easily derive the following results.
\begin{lemma}\label{moment1}
For the operators defined in \eqref{cosnew} we have
\begin{align}
L_n^{\rho}(1;x)=&1 \label{e0},\\
L_n^{\rho}(\rho;x)=&\rho(x)\tanh(n\rho(x))\label{e1},\\
L_n^{\rho}(\rho^2;x)=&\rho^2(x)+\frac{\rho(x)}{n}\tanh(n\rho(x)), \label{e2}\\
L_n^{\rho}(\rho^3;x)=&\rho^3(x)\tanh(n\rho(x))+\frac{3\rho^2(x)}{n}+\frac{\rho(x)}{n}\tanh(n\rho(x))\label{e3},\\
L_n^{\rho}(\rho^4;x)=&\rho^4(x)+\frac{6\rho^3(x)}{n}\tanh(n\rho(x))+7\frac{\rho^2(x)}{n^2}+\frac{\rho(x)}{n^3}\tanh(n\rho(x))\label{e4}.
 \end{align}
\end{lemma}
\begin{lemma}\label{moment2}
For the operators defined in \eqref{cosnew} we have
\begin{align*}
L_n^{\rho}(\rho(t)-\rho(x);x)=&\rho(x)(\tanh(n\rho(x))-1),\\
L_n^{\rho}((\rho(t)-\rho(x))^2;x)=&\left(2\rho^2(x)-\frac{\rho(x)}{n}\right)(1-\tanh(n\rho(x)))+\frac{\rho(x)}{n},\\
%L_n^{\rho}((\rho(t)-\rho(x))^3;x)=&\rho^3(x)\tanh(n\rho(x))+\frac{3\rho^2(x)}{n}+\frac{\rho(x)}{n}\tanh(n\rho(x))\\
L_n^{\rho}((\rho(t)-\rho(x))^4;x)=&\left(8\rho^4(x)-\frac{12\rho^3(x)}{n}+\frac{4\rho^2(x)}{n^2}-\frac{\rho(x)}{n^3}\right)\\&\times(1-\tanh(n\rho(x)))
+\frac{3\rho^2(x)}{n^2}+\frac{\rho(x)}{n^3}.
 \end{align*}
\end{lemma}
Now we give a very useful lemma.
\begin{lemma}\label{moment3}
For the operators defined in \eqref{cosnew} we have
\begin{align*}
\lim_{n \to \infty}nL_n^{\rho}(\rho(t)-\rho(x);x)=0,\\
\lim_{n \to \infty}nL_n^{\rho}((\rho(t)-\rho(x))^2;x)=\rho(x).\\
%L_n^{\rho}((\rho(t)-\rho(x))^3;x)=&\rho^3(x)\tanh(n\rho(x))+\frac{3\rho^2(x)}{n}+\frac{\rho(x)}{n}\tanh(n\rho(x))\\
%L_n^{\rho}((\rho(t)-\rho(x))^4;x)=&\left(8\rho^4(x)-\frac{12\rho^3(x)}{n}+\frac{4\rho^2(x)}{n^2}-\frac{\rho(x)}{n^3}\right)(1-\tanh(n\rho(x)))\\
%&+\frac{3\rho^2(x)}{n^2}+\frac{\rho(x)}{n^3}
 \end{align*}
 \end{lemma}
 \begin{proof}From Lemma \ref{moment2}
\begin{align*}
nL_n^{\rho}(\rho(t)-\rho(x);x)=&n\rho(x)(\tanh(n\rho(x))-1)\\
=&\frac{-2n\rho(x)}{e^{2n\rho(x)}+1}.\\
 \end{align*}
 Thus
 \begin{align*}
\lim_{n \to \infty}nL_n^{\rho}(\rho(t)-\rho(x);x)=&0 .\end{align*} Again from Lemma \ref{moment2}
\begin{align*}
nL_n^{\rho}((\rho(t)-\rho(x))^2;x)=&\left(2\rho(x)-\frac{1}{n}\right)n\rho(x)(1-\tanh(n\rho(x)))+\rho(x)\\
=&\left(2\rho(x)-\frac{1}{n}\right)\left(\frac{2n\rho(x)}{e^{2n\rho(x)}+1}\right)+\rho(x).\\
 \end{align*}
 Thus we have
 $$\lim_{n \to \infty}nL_n^{\rho}((\rho(t)-\rho(x))^2;x)=\rho(x).$$
\end{proof}
We prove the convergence theorem using weighted Korovkin type theorem. Korovkin's theorem \cite{pp} was extended to unbounded intervals and a weighted
Korovkin type theorem in a subspace of continuous functions on the real axis $\R$ was proved in \cite{adg1}, \cite{adg2}. It was shown that the test
functions $1$, $x$, $x^2$ of original Korovkin's theorem  can be replaced by $1,$ $\rho$, $\rho^2$ under certain additional conditions on the function
$\rho$. We recall some notations and results given in \cite{ali2}, \cite{adg1}, \cite{adg2}.

Let $\varphi(x)=1+\rho^2(x)$, where $\rho$ satisfies conditions
$(\rho_1)$ and  $(\rho_2)$. Thus we see that $\rho$ is continuous
and strictly increasing function on positive real axis. We will
consider following weighted space:
$$B_\varphi(\R^+)=\{f:\R^+ \to \R :~ ~|f(x)|\leq M_f\varphi(x),~ ~x \in \R^+\},$$
where $M_f$ is positive constant depending only on $f$. $B_\varphi(\R^+)$ is a normed space with the norm
$$||f||_\varphi = \sup_{x \in \R^+}\frac{|f(x)|}{\varphi(x)}.$$
We denote the subspace of all continuous function in $B_\varphi(\R^+)$ by $C_\varphi(\R^+)$. $C^k_\varphi(\R^+)$ denotes the subspace of all functions $f
\in C_\varphi(\R^+)$ with the property $$\displaystyle \lim_{x \to \infty}\frac{|f(x)|}{\varphi(x) }=k_f,$$ where ${k_f}$ is a constant depending on $f$.
$U_\varphi(\R^+)$ be the subspace of all functions f in $C_\varphi(\R^+) $ such that $\frac{f(x)}{\varphi(x) }$ is uniformly continuous. Then obviously
$$C^k_\varphi(\R^+)\subset U_\varphi(\R^+)\subset C_\varphi(\R^+)\subset
B_\varphi(\R^+).$$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{lemma} [\cite{adg1,adg2}]\label{lem A} \textit{The linear positive operators} $L_n$, $n\geq 1$, \textit{act from} $C_\varphi(\R^+)$ to $ B_\varphi(\R^+)$ \textit{if
and only if}
$$|L_n(\varphi;x)|\leq K\varphi(x),$$
\textit{where} $x \in \R^+$, $\varphi(x)$ \textit{is the weight
function} \textit{and} K \textit{is a positive constant.}\end{lemma}%%%%%%%%%%%%%%%%%%%%
\begin{theorem}[\cite{adg1,adg2}]\label{thm A}\textit{Let} $(L_n)_{n \geq1}$ \textit{be the sequence of positive linear operators which act from} $ C_\varphi(\R^+)$
\textit{to} $B_\varphi(\R^+)$ \textit{satisfying the conditions}
$$\lim_{n \to \infty}||L_n(\rho^i)-\rho^i||_{\varphi}=0,~  ~i=0,1,2.$$
\textit{then for any function} $f \in C^k_\varphi(\R^+)$
$$\lim_{n \to \infty}||L_n(f)-f||_{\varphi}=0.$$\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
The linear positive operators $L_n^{\rho}$, $n\in \N$, act from $C_\varphi(\R^+)$ to $ B_\varphi(\R^+)$, where $\varphi(x)=1+\rho^2(x)$ is the weight
function.
\end{lemma}
\begin{proof}
In view of \eqref{e0} and \eqref{e2} we see that operators $L_n^{\rho}$, $n\in \N$ satisfy the condition of the Lemma \ref{lem A}. Thus the result follows.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In \cite{lrms2} the following inequality was proved
$$0\leq x^r(1-tanh(nx)) \leq 2^{1-r} r! n^{-r},~ ~n,r\in
N~ ~and~ ~x\geq 0 .$$ Similarly for $\rho(x)$ satisfying $\rho_1$ and  $\rho_2 $  and $ n,r\in N$ we can get the following inequality
 \begin{equation}\label{tanh}
 0\leq \rho^r(x)(1-tanh(n\rho(x))
\leq 2^{1-r} r! n^{-r}. \end{equation} Now we prove the convergence theorem for the operators $(L_n^{\rho})$.
\begin{theorem}
Let $(L_n^{\rho})_{n\in \N}$ be the sequence of linear positive operators defined by \eqref{cosnew}. Then for any $f \in C^k_\varphi(\R^+)$ we have
$$\lim_{n \to \infty}||L_n^{\rho}(f)-f||_{\varphi}=0.$$
\end{theorem}
\begin{proof}Using Theorem \ref{thm A} we see that in order to prove the theorem, it is sufficient to prove
the following three conditions
$$\lim_{n \to \infty}||L_n^{\rho}(\rho^v)-\rho^v||_{\varphi}=0,~   ~v
=0,1,2.$$

 Now from \eqref{e0} we have
$$\lim_{n \to \infty}||L_n^{\rho}(1)-1||_{\varphi}=0.$$

From \eqref{e1} we get
$$||L_n^{\rho}(\rho)-\rho||_{\varphi}\leq\sup_{x \in \R^+}\frac{\rho(x)}{1+\rho^2(x)}(1-\tanh(n\rho(x)),$$
so using \eqref{tanh} for $r=1$ we have
$$||L_n^{\rho}(\rho)-\rho||_{\varphi}\leq\frac{1}{n}.$$

This leads to
$$\lim_{n \to \infty}||L_n^{\rho}(\rho)-\rho||_{\varphi}=0.$$

Again from \eqref{e2}
\begin{align*}
L_n^{\rho}(\rho^2)-\rho^2=&\frac{\rho(x)}{n}\tanh(n\rho(x))\\
=&\frac{\rho(x)}{n}-\frac{\rho(x)}{n}(1-\tanh(n\rho(x))),
 \end{align*}
 thus
\begin{align*}
||L_n^{\rho}(\rho^2)-\rho^2||_{\varphi}\leq &\sup_{x \in \R^+}\Big[\frac{\rho(x)}{n(1+\rho^2(x))}+\frac{\rho(x)}{n(1+\rho^2(x))}(1-\tanh(n\rho(x)))\Big]
 \end{align*}
 and using \eqref{tanh} we get
 \begin{align}\label{rho2}
||L_n^{\rho}(\rho^2)-\rho^2||_{\varphi}\leq&\Big[\frac{1}{n}+\frac{1}{n}\Big]=\frac{2}{n}.
 \end{align}
 So we have
 $$\lim_{n \to \infty}||L_n^{\rho}(\rho^2)-\rho^2||_{\varphi}=0.$$
 This completes the proof.
 \end{proof}
 \section{Rate of Convergence via Weighted Modulus of Continuity}
In this section we compute the rate of convergence of the
operators defined in \eqref{cosnew} in terms of weighted modulus
of continuity. In \cite{holhos} Holho\c{s} defined for all $ f \in
C_\varphi(\R^+)$ and for every $\delta \geq 0$, the weighted
modulus of continuity as
$$\omega_\rho(f,\delta)=\sup_{x \in \R^+,~ ~|\rho(t)-\rho(x)|\leq \delta}\frac{|f(t)-f(x)|}{\varphi(t)+\varphi(x)}.$$
%Following properties of weighted modulus of continuity were also
%shown in \cite{holhos} (i)\\
We see that $\omega_\rho(f,0)=0$ for all $ f \in C_\varphi(\R^+)$ and also that $\omega_\rho(f,\delta)$ is a nonnegative and nondecreasing function with
respect to $\delta$. The properties of weighted modulus of continuity were discussed in \cite{holhos}. The following results were given by Holhos
\cite{holhos}.
\begin{lemma}[\cite{holhos}]\label{lem B}\textit{ For every} $ f \in U_\varphi(\R^+)$, $\displaystyle \lim_{\delta \to 0}\omega_\rho(f,\delta)=0$. \end{lemma}
\begin{theorem}[\cite{holhos}]\label{thm B}\textit{Let } $(L_n)_{n\geq 1}$ \textit{ be a sequence of linear positive operators} \textit{acting from} $C_\varphi(\R^+)$ to $
B_\varphi(\R^+)$ \textit{with}
\begin{align*}
||L_n(\rho^0)-\rho^0||_{\varphi^0}=&a_n,\\
||L_n(\rho)-\rho||_{\varphi^\frac{1}{2}}=&b_n,\\
||L_n(\rho^2)-\rho^2||_{\varphi}=&c_n,\\
||L_n(\rho^3)-\rho^3||_{\varphi^\frac{3}{2}}=&d_n,
 \end{align*}
 where $a_n$, $b_n,$ $c_n$ and $d_n$ tend to zero as $n$ goes to infinity.
 Then
$$||L_n(f)-f||_{\varphi^\frac{3}{2}}\leq(7+4a_n+2c_n)\omega_\rho(f,\delta_n)+||f||_{\varphi}a_n$$
for all $f \in C_\varphi(\R^+)$, where
$$\delta_n=2\sqrt{(a_n+2b_n+c_n)(1+a_n)}+a_n+3b_n+3c_n+d_n.$$\end{theorem}
\begin{theorem}\label{rate}
For all $ f \in C_\varphi(\R^+)$, we have
$$||L_n^{\rho}(f)-f||_{\varphi^\frac{3}{2}}\leq\left(7+\frac{4}{n}\right)\omega_\rho(f,\delta_n),$$
where
$$\delta_n=\frac{4}{\sqrt{n}}+\frac{15}{n}.$$
\end{theorem}
\begin{proof}
From \eqref{e0} and \eqref{e1} we see that
$$||L_n(\rho^0)-\rho^0||_{\varphi^0}=a_n=0,$$
\begin{align*}
b_n=&||L_n(\rho)-\rho||_{\varphi^\frac{1}{2}}\leq\sup_{x \in \R^+}\frac{\rho(x)}{\sqrt{1+\rho^2(x)}}(1-\tanh(n\rho(x))
 \end{align*}
 and using \eqref{tanh} we get
 $$b_n=||L_n(\rho)-\rho||_{\varphi^\frac{1}{2}}\leq\frac{1}{n}.$$
From \eqref{rho2} we have
$$c_n=||L_n(\rho^2)-\rho^2||_{\varphi}\leq\frac{2}{n}.$$
Again from \eqref{e3}  we obtain
\begin{align*}
d_n=&||L_n(\rho^3)-\rho^3||_{\varphi^\frac{3}{2}}\\
=&\sup_{x \in \R^+}\frac{1}{(1+\rho^2(x))^\frac{3}{2}}\Big|\rho^3(x)\tanh(n\rho(x))-\rho^3(x)+\frac{3\rho^2(x)}{n}\\
&+\frac{\rho(x)}{n^2}\tanh(n\rho(x))\Big|\\
\leq&\sup_{x \in \R^+}\frac{1}{(1+\rho^2(x))^\frac{3}{2}}\Big|\rho^3(x)(1-\tanh(n\rho(x)))+\frac{\rho(x)}{n^2}(1-\tanh(n\rho(x)))\\
&+\frac{3\rho^2(x)}{n}+\frac{\rho(x)}{n^2}\Big|\\
\leq&\frac{ 1}{n}+\frac{ 1}{n^2}+\frac{ 3}{n}+\frac{ 1}{n^2}.
 \end{align*}
Using \eqref{tanh} and by the fact that $\frac{1}{n^2}\leq\frac{1}{n}$ we obtain
\begin{align*}
d_n=&||L_n(\rho^3)-\rho^3||_{\varphi^\frac{3}{2}} \leq \frac{ 6}{n}.
 \end{align*}
Thus we see that $a_n$, $b_n$, $c_n$ and $d_n$ tend to zero as $n$ goes to infinity. So on applying Theorem \ref{thm B}, we get
$$||L_n^{\rho}(f)-f||_{\varphi^\frac{3}{2}}\leq\Big(7+\frac{4}{n}\Big)\omega_\rho(f,\delta_n),$$
 where
$$\delta_n=\frac{4}{\sqrt{n}}+\frac{15}{n}.$$
This completes the proof.
\end{proof}
\begin{remark}
We see from Theorem \ref{rate} that as $n \to \infty$, $\delta_n \to 0.$ Thus, using Lemma \ref{lem B}, we have
$$\lim_{n \to \infty}||L_n^{\rho}(f)-f||_{\varphi^\frac{3}{2}}=0$$
 for every $ f \in U_\varphi(\R^+)$.
 \end{remark}
\section{Voronovskaja Type Theorem}
Now we give a Voronovskaja-type result using the technique of C\'{a}rdenas-Morales et al. \cite{cardens}.
\begin{theorem}
Let $ f \in C_\varphi(\R^+),$ $x \in \R^+$ and suppose that the first and second derivatives of $f \circ \rho^{-1}$ exist at $\rho(x)$. If the second
derivative of $f \circ \rho^{-1}$ is bounded on $\R^+$, then we have
$$\lim_{n \to \infty}n[L_n^{\rho}(f;x)-f(x)]=\frac{\rho(x)}{2}(f \circ \rho^{-1})''(\rho(x)).$$
\end{theorem}
\begin{proof}
By the Taylor expansion of $f \circ \rho^{-1}$ at the point $\rho(x) \in \R^+$, there exists $\xi$ lying between $x$ and $t$ such that
\begin{align*}\label{taylor}
f(t) = &(f \circ \rho^{-1})(\rho(t))=(f \circ \rho^{-1})(\rho(x)) + (f \circ \rho^{-1})'(\rho(x))(\rho(t)-\rho(x)) \\&+ \frac{1}{2}(f \circ
\rho^{-1})''(\rho(x))(\rho(t)-\rho(x))^2 + h(t;x)(\rho(t)-\rho(x))^2,
\end{align*}
where
\begin{equation}
h(t;x)=\frac{(f \circ \rho^{-1})''(\rho(\xi))-(f \circ \rho^{-1})''(\rho(x))}{2}.
\end{equation}
On applying the operator \eqref{cosnew}
\begin{align}\label{voro1}
n[L_n^{\rho}&(f;x)-f(x)]\nonumber\\=&\nonumber(f \circ \rho^{-1})'(\rho(x))nL_n^{\rho}(\rho(t)-\rho(x);x) + \frac{1}{2}(f \circ
\rho^{-1})''(\rho(x))\\&\times nL_n^{\rho}((\rho(t)-\rho(x))^2;x)+n L_n^{\rho}(h(t;x)(\rho(t)-\rho(x))^2;x).
\end{align}
Now using Lemma \ref{moment3} in \eqref{voro1} we get
\begin{align}\label{voro2}
\lim_{n \to \infty}n[L_n^{\rho}&(f;x)-f(x)]\nonumber\\=&\frac{\rho(x)}{2}(f \circ \rho^{-1})''(\rho(x)) +\lim_{n \to \infty}n
L_n^{\rho}(h(t;x)(\rho(t)-\rho(x))^2;x).
\end{align}
From the hypothesis of the theorem we have $|h(t;x)|\leq M$ and $$\displaystyle \lim_{t \to x}h(t;x)=0.$$ Thus, for any $\varepsilon >0$ there exist
 a $\delta >0$ such that
$$|h(t;x)|< \varepsilon ~ for ~ |t-x| <
 \delta.$$
 But from the condition $(\rho_2)$ we have
 $$|t-x|\leq|\rho(t)-\rho(x)|.$$
 Therefore, if $|\rho(t)-\rho(x)| < \delta$, then
 $$|h(t;x)(\rho(t)-\rho(x))^2|<\varepsilon (\rho(t)-\rho(x))^2$$and
 if  $$|\rho(t)-\rho(x)| \geq \delta,$$ then $$|h(t;x)(\rho(t)-\rho(x))^2|<\frac{M}{\delta^2}
 (\rho(t)-\rho(x))^4.$$

 Hence
\begin{align*}L_n^{\rho}(h(t;x)&(\rho(t)-\rho(x))^2;x) \\ &<\varepsilon L_n^{\rho}((\rho(t)-\rho(x))^2;x)+\frac{M}{\delta^2} L_n^{\rho}((\rho(t)-\rho(x))^4;x).\end{align*}
 From
Lemma \ref{moment2} we see that
$$L_n^{\rho}((\rho(t)-\rho(x))^4;x)=O\Big(\frac{1}{n^2}\Big).$$
Thus we get
$$\lim_{n \to \infty}nL_n^{\rho}(h(t;x)(\rho(t)-\rho(x))^2;x)=0.$$
On applying this to \eqref{voro2} we get the desired result.
\end{proof}
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