\documentclass[twoside,leqno,%draft
]{factaMI}
\usepackage{latexsym,amssymb,amsfonts,euscript}
\usepackage[section]{placeins}
\usepackage[dvips]{graphicx}
\input epsf
%\usepackage{rotating}

\def\navod#1{$\,{#1}^\circ$}
\def\dfrac#1#2{{\displaystyle {{#1}}\over{\displaystyle{#2}}}}
\def\qfrac#1#2{\left[#1\atop #2\right]}
\def\DS{\displaystyle}


 \setcounter{page}{1}  %%% will be entered by the editor


 %%% Insert short name of all Authors
 %%% and short title for header
 \markboth{Vijay Gupta and Ali Aral}  %%% short names
 {Bernstein Durrmeyer Operators Based on Two Parameters}                       %%% short (or whole) title
                                                  %%% (capital initials only)

 \factanis{22\,(2007),  -- }   %%% will be entered by the editor
 \Received{June 20, 2007.}     %%% will be entered by the editor

 %%% Insert Primary and Secondary
 %%% 2000 Mathematics Subject Classification numbers
 \AMSMSC{Primary 41A25; Secondary 41A30}

 %%% Insert title (all capital letters)
 %%% Also grants or other notes
 %%% about the article that should go on the front page should be placed here.
\title{Bernstein Durrmeyer Operators Based on Two Parameters
}

 %%% Inser full names of all Authors
\author{Vijay Gupta and Ali Aral}

\begin{document}

\maketitle


 %%% Insert a brief summary (50-150 words)
\abstract
 In the present paper, we study the
applications of the extension of quantum calculus based on two parameters. We
define beta function and establish an identity with gamma function, for two parameters $(p,q)$, i e. the post-quantum
calculus. We also propose the $(p,q)$-Durrmeyer operators, estimate moments
and establish some direct results. Depending on the selection of $p$ and $q,$
the rate of convergence of the our new operators can provide better approximation than those of
the Bernstein-Durrmeyer operators and its $q$-analogue. In the end, we provide some graphs using the software Mathematica.
\endabstract



\section{Introduction}\label{sec:section1}
 \vglue-10pt
 \indent

In the last eighteen years the applications of quantum calculus ($q$%
-calculus) in the field of approximation theory is an active area of
research. Several new operators have been generalized to their $q$ variants
and their approximation behavior have been discussed, we mention here the
recent books \cite{avr} and \cite{vgrpa} on this topic. The further
generalization of $q$ calculus is the post quantum calculus, denoted by $%
(p,q)$-calculus. Some papers have appeared recently in this area related to
approximation theory. This area is in developing stage. For $(p,q)$%
-calculus, we mention here some basic definitions and theorems, some of them
may be found in the recent papers \cite{1}, \cite{3}, \cite{4}, \cite{VSSY} etc.\\

The $\left( p,q\right)$-number is defined as
$$
\left[ n\right] _{p,q}=\frac{p^{n}-q^{n}}{p-q},
$$
Obviously, it may be seen that $\left[ n\right] _{p,q}=p^{n-1}\left[ n\right]
_{q/p}.$ It is obvious from the definition, that $q$-integers and $(p,q)$%
-integers are different, that is we cannot obtain $(p,q)$ integers just by
replacing $q$ by $q/p$ in the definition of $q$-integers. But if we put $p=1$
in definition of $(p,q)$ integers then $q$-integers becomes a particular
case of $(p,q)$ integers. Thus we can say that $(p,q)$-integers can be taken
as a generalization of $q$-integers. The $\left( p,q\right) $-factorial is
defined by
$$
\left[ n\right] _{p,q}!=\prod_{k=1}^{n}\left[ k\right] _{p,q},n\geq 1,
\quad \left[0\right] _{p,q}!=1.
$$
The $\left( p,q\right) $-binomial coefficient is given by%
$$
\qfrac{n}{k}_{p,q}=\frac{\left[ n\right] _{p,q}!}{\left[ n-k\right] _{p,q}!\left[
k\right] _{p,q}!},\quad 0\leq k\leq n.
$$

\begin{definition}
The $(p,q)$-power basis is defined below
\begin{eqnarray*}
(x\ominus a)_{p,q}^{n}&=&(x-a)(px-qa)(p^2x-q^2a)\cdots (p^{n-1}x-q^{n-1}a).
\end{eqnarray*}
\end{definition}

\begin{definition}
The $(p,q)$-derivative of the function $f$ is defined as
$$
D_{p,q}f\left( x\right) =\frac{f\left( px\right) -f\left( qx\right) }{\left(
p-q\right) x},\quad x\neq 0
$$
and $D_{p,q}f\left( 0\right) =f^{^{\prime }}\left( 0\right) $, provided that
$f$ is differentiable at $0$. It is obvious that $D_{p,q}x^{n}=\left[ n%
\right] _{p,q}x^{n-1}$. Note also that for $p=1$, the $(p,q)$-derivative
reduces to the $q-$derivative given by%
$$
D_{q}f\left( x\right) =\frac{f\left( x\right) -f\left( qx\right) }{\left(
1-q\right) x},\quad x\neq 0
$$
\end{definition}

\begin{definition}
The $(p,q)$-derivative fulfils the following product rules%
\begin{eqnarray*}
D_{p,q}(f(x)g(x)) &=&f(px)D_{p,q}g(x)+g(qx)D_{p,q}f(x) \\[2mm]
D_{p,q}(f(x)g(x)) &=&g(px)D_{p,q}f(x)+f(qx)D_{p,q}g(x).
\end{eqnarray*}
\end{definition}

The following assertion is valid:%
\begin{eqnarray*}
D_{p,q}(x\ominus a)_{p,q}^{n} &=&\left[ n\right] _{p,q}(px\ominus
a)_{p,q}^{n-1},n\geq 1 \\[2mm]
D_{p,q}(a\ominus x)_{p,q}^{n} &=&-\left[ n\right] _{p,q}(a\ominus
qx)_{p,q}^{n-1},n\geq 1,
\end{eqnarray*}%
and $D_{p,q}(x\ominus a)_{p,q}^{0}=0.$

\begin{definition}
Let $f$ be an arbitrary function and $a$ be a real number. The $(p,q)$%
-integral of $f(x)$ on \ $\left[ 0,a\right] $ is defined as
$$
\int_{0}^{a}f\left( x\right) d_{p,q}x=\left( q-p\right) a\sum
\limits_{k=0}^{\infty }\frac{p^{k}}{q^{k+1}}f\left( \frac{p^{k}}{q^{k+1}}%
a\right) \  \  \mbox{ if } \  \  \left \vert \frac{p}{q}\right \vert <1
$$
and
$$
\int_{0}^{a}f\left( x\right) d_{p,q}x=\left( p-q\right) a\sum
\limits_{k=0}^{\infty }\frac{q^{k}}{p^{k+1}}f\left( \frac{q^{k}}{p^{k+1}}%
a\right) \  \  \mbox{ if } \  \  \left \vert \frac{p}{q}\right \vert >1
$$
\end{definition}

\begin{theorem}
(Fundamental theorem of $(p,q)$-calculus) If $F(x)$ is an antiderivative of $%
f(x)$ and $F(x)$ is continuous at $x=0$, we have
$$\int \nolimits_{a}^{b}f\left( x\right) d_{p,q}x=F(b)-F(a),$$
where $0\leq a<b\le \infty .$
\end{theorem}

\begin{proposition}
The formula of $(p,q)$-integration by part is given by
\begin{equation}\label{a1}
\ \ \int_{a}^{b}f ( px ) D_{p,q}g ( x ) d_{p,q}x=f(
b) g( b) -f( a) g ( a )-\int
_{a}^{b}g ( qx ) D_{p,q}f ( x ) d_{p,q}x.
\end{equation}
\end{proposition}

\begin{definition}
 (\cite{2})  Let $n$ is a nonnegative integer, we define the $(p,q)$-Gamma
function as%
$$\Gamma _{p,q}\left( n+1\right) =\frac{(p\ominus q)_{p,q}^{n}}{(p-q)^{n}}=%
\left[ n\right] _{p,q}!, \  \ 0<q<p.$$
\end{definition}

In the present paper, we define $(p,q)$-Beta function and establish a
relation between $(p,q)$-Beta and $(p,q)$-Gamma functions. As the $(p,q)$%
-Beta function is generalization of Beta function of first kind, we propose
the $(p,q)$-Bernstein-Durrmeyer operators. Using some identities of $(p,q)$%
-calculus, we estimate moments and establish some direct results for $(p,q)$%
-Bernstein-Durrmeyer operators.

\section{$(p,q)$-Beta Function}\label{sec:section2}
 \vglue-10pt
 \indent

In this section, we propose the $(p,q)$ Beta function and then find a
relation between $(p,q)$-Beta and $(p,q)$-Gamma functions.

Let $m,n\in \mathbb{N},$ we define $\left( p,q\right) $-Beta function of first kind as%
\begin{equation}
B_{p,q}\left( m,n\right) =\int \nolimits_{0}^{1}\left( px\right)
^{m-1}\left( p\ominus pqx\right) _{p,q}^{n-1}d_{p,q}x.  \label{a5}
\end{equation}


In \cite{2}, the author has given the relation between $(p,q)$-Beta and $(p,q)$-Gamma functions, without mentioning the integral representation of the $(p,q)$-Beta functions. For the above form of $(p,q)$-Beta function, we have the following relation, which is not commutative.
\begin{theorem}
The $(p,q)$-Gamma and $(p,q)$-Beta functions fulfill the following
fundamental relation%
\begin{equation}
B_{p,q}\left( m,n\right) =p^{[n\left( 2m+n-2\right)+n-2] /2}\frac{\Gamma
_{p,q}\left( m\right) \Gamma _{p,q}\left( n\right) }{\Gamma _{p,q}\left(
m+n\right) },  \label{a4}
\end{equation}%
where $m,n\in \mathbb{N}.$
\end{theorem}

\begin{proof}
For any $m,n\in $ $\mathbb{N}$ since%
$$B_{p,q}\left( m,n\right) =\int \nolimits_{0}^{1}\left( px\right)
^{m-1}\left( p\ominus pqx\right) _{p,q}^{n-1}d_{p,q}x,$$
using (\ref{a1}) for $f\left( x\right) =x^{m-1}$ and $g\left( x\right) =-%
\frac{\left( p\ominus px\right) _{p,q}^{n}}{p\left[ n\right] _{p,q}}$ with
the equality 
$$D_{p,q}\left( p\ominus px\right) ^{n}=-\left[ n\right]
_{p,q}p\left( p\ominus pqx\right) ^{n-1}$$ 
we have
\begin{eqnarray}
B_{p,q}\left( m,n\right) &=&\frac{\left[ m-1\right] _{p,q}}{p^{m-1}\left[ n%
\right] _{p,q}}\int \nolimits_{0}^{1}\left( px\right) ^{m-2}\left( p\ominus
pqx\right) _{p,q}^{n}d_{p,q}x  \nonumber \\
&=&\frac{\left[ m-1\right] _{p,q}}{p^{m-1}\left[ n\right] _{p,q}}%
B_{p,q}\left( m-1,n+1\right).  \label{a2}
\end{eqnarray}%
Also we can write for positive integer $n$%
\begin{eqnarray*}
B_{p,q}\left( m,n+1\right) &=&\int \nolimits_{0}^{1}\left( px\right)
^{m-1}\left( p\ominus pqx\right) _{p,q}^{n}d_{p,q}x \\
&=&\int \nolimits_{0}^{1}\left( px\right) ^{m-1}\left( p\ominus pqx\right)
_{p,q}^{n-1}\left( p^{n}-pq^{n}x\right) d_{p,q}x \\
&=&p^{n}\int \nolimits_{0}^{1} ( px ) ^{m-1} ( p\ominus
pqx ) _{p,q}^{n-1}d_{p,q}x \\
&&\qquad -q^{n}\int \nolimits_{0}^{1}\left( px\right) ^{m}\left( p\ominus
pqx\right) _{p,q}^{n-1}d_{p,q}x \\
&=&p^{n}B_{p,q}\left( m,n\right) -q^{n}B_{p,q}\left( m+1,n\right) .
\end{eqnarray*}%
Using (\ref{a2}), we have
$$
B_{p,q}\left( m,n+1\right) =p^{n}B_{p,q}\left( m,n\right) -q^{n}\frac{\left[
m\right] _{p,q}}{p^{m}\left[ n\right] _{p,q}}B_{p,q}\left( m,n+1\right),
$$
which implies that
$$B_{p,q}\left( m,n+1\right) =p^{n+m}\frac{p^{n}-q^{n}}{p^{n+m}-q^{n+m}}%
B_{p,q}\left( m,n\right) .
$$
Further, by definition of $(p,q)$ integration
$$B_{p,q}\left( m,1\right) =\int \nolimits_{0}^{1}\left( px\right)
^{m-1}d_{p,q}x=\frac{p^{m-1}}{\left[ m\right] _{p,q}}.
$$
We immediately have%
\begin{eqnarray*} 
&&B_{p,q}\left( m,n\right)=p^{n+m-1}\frac{p^{n-1}-q^{n-1}}{ 
p^{n+m-1}-q^{n+m-1}}B_{p,q}\left( m,n-1\right)\qquad \\
&&\qquad=p^{n+m-1}\frac{p^{n-1}-q^{n-1}}{p^{n+m-1}-q^{n+m-1}}p^{n+m-2}\frac{%
p^{n-2}-q^{n-2}}{p^{n+m-2}-q^{n+m-2}}B_{p,q}\left( m,n-2\right)\\
&&\qquad=p^{n+m-1}\frac{p^{n-1}-q^{n-1}}{p^{n+m-1}-q^{n+m-1}}p^{n+m-2}\frac{%
p^{n-2}-q^{n-2}}{p^{n+m-2}-q^{n+m-2}}\cdots \\
&&\qquad\quad\cdots p^{m+1}\frac{p-q}{p^{m+1}-q^{m+1}}%
B_{p,q}\left( m,1\right) \\
&&\qquad=p^{n+m-1}\frac{p^{n-1}-q^{n-1}}{p^{n+m-1}-q^{n+m-1}}p^{n+m-2}\frac{%
p^{n-2}-q^{n-2}}{p^{n+m-2}-q^{n+m-2}}\cdots\\
&&\qquad\quad\cdots p^{m+1}\frac{p-q}{p^{m+1}-q^{m+1}}%
\frac{p^{m-1}}{\left[ m\right] _{p,q}}\\
&&\qquad=\frac{p^{(m-1)+m+(m+1)+\cdots+(m+n-1)}}{p^m}\frac{\left( p\ominus q\right)
_{p,q}^{n-1}}{\left( p^{m}\ominus q^{m}\right) _{p,q}^{n}}\left( p-q\right),
\end{eqnarray*}
i.e.,
\begin{equation}
B_{p,q}\left( m,n\right)
=p^s\frac{\left( p\ominus q\right)
_{p,q}^{n-1}}{\left( p^{m}\ominus q^{m}\right) _{p,q}^{n}}\left( p-q\right),
\label{a3}	
\end{equation}
where $s={[n\left( 2m+n-2\right)+n-2] /2}$.


Following \cite{2}, we have $(a\ominus b)_{p,q}^{n+m}=(a\ominus
b)_{p,q}^n(ap^n\ominus bq^n)_{p,q}^m$ thus (\ref{a3}) leads to
\begin{eqnarray*}
B_{p,q}\left( m,n\right) &=&p^s\frac{\left(
p\ominus q\right) _{p,q}^{n-1}}{\left( p^{m}\ominus q^{m}\right) _{p,q}^{n}}%
\left( p-q\right) \\
&=&p^s\frac{\left( p\ominus q\right)
_{p,q}^{n-1}}{\left( p-q\right) ^{n-1}}.\frac{\left( p\ominus q\right)
_{p,q}^{m-1}}{\left( p-q\right) ^{m-1}}.\frac{\left( p-q\right) ^{m-1}\left(
p-q\right) ^{n-1}}{\left( p\ominus q\right) _{p,q}^{m-1}\left( p^{m}\ominus
q^{m}\right) _{p,q}^{n}}\left( p-q\right) \\
&=&p^s\frac{\left( p\ominus q\right)
_{p,q}^{n-1}}{\left( p-q\right) ^{n-1}}.\frac{\left( p\ominus q\right)
_{p,q}^{m-1}}{\left( p-q\right) ^{m-1}}.\frac{\left( p-q\right) ^{m+n-1}}{%
\left( p\ominus q\right) _{p,q}^{m+n-1}} =p^s\frac{\Gamma _{p,q}\left( m\right)
\Gamma _{p,q}\left( n\right) }{\Gamma _{p,q}\left( m+n\right) }.
\end{eqnarray*}
This completes the proof of the theorem.
\end{proof}

\section{$(p,q)$ Bernstein-Durrmeyer-Operators and Moments}\label{sec:section3}
 \vglue-10pt
 \indent

The $\left( p,q\right) $-analogue of Bernstein operators for $x\in \left[ 0,1%
\right]$ and $0<q<p\le 1$ can be defined as
$$
B_{n,p,q}\left( f;x\right) =\sum_{k=0}^{n}b_{n,k}^{p,q}(1,x)f\left(\frac{%
p^{n-k}[k]_{p,q}}{[n]_{p,q}}\right) ,
$$
where



$$
b_{n,k}^{p,q}(1,x)=\qfrac{n}{k}_{p,q}p^{[k(k-1)-n(n-1)]/2} x^{k}(1\ominus x)_{p,q}^{n-k} .
$$
Although the different forms of the $(p,q)$-Bernstein polynomials have been
considered in \cite{1} and \cite{4}, but due to technical problems, the
forms considered in these two papers do not preserve even the constant
functions.

\begin{remark}
\label{r1} The link between $q$-binomial coefficient and $(p,q)$-binomial
coefficients can be described as:
\begin{eqnarray*}
\qfrac{n}{k}_{q/p}&=&\frac{[n]_{q/p}!}{[k]_{q/p}![n-k]_{q/p}!} \\[2mm]
&=&\frac{[n]_{q/p}[n-1]_{q/p}\cdots [2]_{q/p}.1}{([k]_{q/p}[k-1]_{q/p}\cdots
[2]_{q/p}.1)([n-k]_{q/p}[n-k-1]_{q/p}\cdots [2]_{q/p}.1)} \\[2mm]
&=&\frac{p^{k(k-1)/2}p^{(n-k)(n-k-1)/2}[n]_{p,q}[n-1]_{p,q}\cdots [2]_{p,q}.1%
}{p^{n(n-1)/2}([k]_{p,q}[k-1]_{p,q}\cdots
[2]_{p,q}.1)([n-k]_{p,q}[n-k-1]_{p,q}\cdots [2]_{p,q}.1)} \\[2mm]
&=&\frac{p^{k(k-1)/2-n(n-1)/2}p^{(n-k)(n-k-1)/2}[n]_{p,q}!}{%
[k]_{p,q}![n-k]_{p,q}!}=p^{k(k-n)}\qfrac{n}{k}_{p,q}.
\end{eqnarray*}
Also, we have $(p,q)$-power basis as
\begin{eqnarray*}
(x\ominus a)_{p,q}^{n}&=&(x-a)(px-qa)(p^2x-q^2a)\cdots (p^{n-1}x-q^{n-1}a) \\
&=&1\cdot p\cdot p^2\cdots p^{n-1}(x-a)\left(x-a\frac{q}{p}\right)\left(x-a\frac{q^2}{%
p^2}\right)\cdots \left(x-a\frac{q^{n-1}}{p^{n-1}}\right) \\
&=&p^{n(n-1)/2}(x-a)_{q/p}^n.
\end{eqnarray*}
Using the above identities and using the moments of $q$-Bernstein
polynomial, it can easily be verified by simple computation that
$$
B_{n,p,q}\left( 1;x\right)=1,\ \ B_{n,p,q}\left( t;x\right)=x,\ \ B_{n,p,q}\left(
t^2;x\right)=x^2+\frac{p^{n-1}x(1-x)}{[n]_{p,q}}.
$$
\end{remark}

The $q$-Durrmeyer operators were proposed in \cite{amc} and also studied in
\cite{ZFVG}. We introduce below the $\left( p,q\right) $ variant of the well
known Durrmeyer operators.

\begin{definition}
The $\left( p,q\right) $-analogue of Bernstein-Durrmeyer operator for $x\in[%
0,1]$ and $0<q<p\le 1$ is defined as
$$
D_{n}^{p,q}(f;x)=\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)%
\int_{0}^{1}b_{n,k}^{p,q}(p,pqt)f\left( t\right) d_{p,q}t
$$%
where $\DS b_{n,k}^{p,q}(p,pqt)=\qfrac{n}{k}_{p,q}\left( pt\right) ^{k}(p\ominus pqt)_{p,q}^{n-k}.$
\end{definition}

It may be remarked here that for $p=1$ these operators will not reduce to
the $q$-Durrmeyer operators, but for $p=q=1$ these will reduce to the
Durrmeyer operators.

\begin{lemma}
\label{l1} For $x\in \lbrack 0,1]$, $0<q<p\leq 1$, we have

\navod1 $D_{n}^{p,q}(1;x)=1$,
\smallskip

\navod2 $D_{n}^{p,q}(t;x)=\frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}$,
\smallskip

\navod3 
$D_{n}^{p,q}(t^2;x)=\frac{p^{2n}[2]_{p,q}}{[n+2]_{p,q}[n+3]_{p,q}} + 
\frac{(2q^2+qp)p^n[n]_{p,q}x}{[n+2]_{p,q}[n+3]_{p,q}} +\frac{
q^3[n]_{p,q}[x^2[n]_{p,q}+p^{n-1}x(1-x)]}{[n+2]_{p,q}[n+3]_{p,q}}$.
\end{lemma}

%\begin{proof}
\noindent {\it Proof.}
Using (\ref{a4}) and (\ref{a5}) and Remark \ref{r1}, we have
{\small
\begin{eqnarray*}
D_{n}^{p,q}(1;x)\!\!\! &=&\!\!\!\!\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)\int_{0}^{1}\qfrac{n}{k}_{p,q}(pt)^{k}(p\ominus pqt)_{p,q}^{n-k}d_{p,q}t \\
\!\!\! &=&\!\!\!\!\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)\qfrac{n}{k}B_{p,q}(k+1,n-k+1) \\
\!\!\! &=&\!\!\!\!\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)\qfrac{n}{k}_{p,q}p^{[n^2+3n-k^2-k]/2}\frac{[k]_{p,q}![n-k]_{p,q}!}{[n+1]_{p,q}!%
} \\
\!\!\! &=&\!\!\!\!B_{n,p,q}\left( 1;x\right) =1.
\end{eqnarray*}}
Next using the identity $[k+1]_{p,q}=p^k+q[k]_{p,q}$ and applying Remark \ref%
{r1}, we have
{\small
\begin{eqnarray*}
D_{n}^{p,q}(t;x)\!\!\!\! &=&\!\!\!\!\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)p^{-1}\int_{0}^{1}\qfrac{n}{k}_{p,q}(pt)^{k+1}(p\ominus pqt)_{p,q}^{n-k}d_{p,q}t \\
\!\!\! &=&\!\!\!\!\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)\qfrac{n}{k}_{p,q}p^{-1} B_{p,q}(k+2,n-k+1) \\
\!\!\! &=&\!\!\!\!\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)\qfrac{n}{k}_{p,q} p^{(n^2+5n-k^2-3k)/2}\frac{[k+1]_{p,q}![n-k]_{p,q}!}{ 
[n+2]_{p,q}!} \\
\!\!\! &=&\!\!\!\!\sum_{k=0}^{n}p^{n-k}b_{n,k}^{p,q}(1,x)\frac{[k+1]_{p,q}}{[n+2]_{p,q}} \\
\!\!\! &=&\!\!\!\!\frac{1}{[n+2]_{p,q}} 
\sum_{k=0}^{n}p^{n-k}b_{n,k}^{p,q}(1,x)(p^{k}+q[k]_{p,q}) \\
\!\!\! &=&\!\!\!\!\frac{p^{n}}{[n+2]_{p,q}}\sum_{k=0}^{n}b_{n,k}^{p,q}(1,x)+\frac{q[n]_{p,q} 
}{[n+2]_{p,q}}\sum_{k=0}^{n}b_{n,k}^{p,q}(1,x)p^{n-k}\frac{[k]_{p,q}}{ 
[n]_{p,q}} \\
\!\!\! &=&\!\!\!\!\frac{1}{[n+2]_{p,q}}p^{n}B_{n,p,q}\left( 1;x\right) +\frac{q[n]_{p,q}}{
[n+2]_{p,q}}B_{n,p,q}\left( t;x\right)\\
\!\!\! &=&\!\!\!\!\frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}.
\end{eqnarray*}}

Further using the identity $[k+2]_{p,q}=p^{k+1}+qp^k+q^2[k]_{p,q}$ and by
Remark \ref{r1}, we get
{\small 
\begin{eqnarray*}
D_{n}^{p,q}(t^2;x)\!\!\!\! &=&\!\!\!\!\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)p^{-2}\int_{0}^{1}\qfrac{n}{k}_{p,q}(pt)^{k+2}(p\ominus pqt)_{p,q}^{n-k}d_{p,q}t \\
\!\!\!\! &=&\!\!\!\!\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)\qfrac{n}{k}_{p,q}p^{-2} B_{p,q}(k+3,n-k+1) \\
\!\!\!\! &=&\!\!\!\!\left[ n+1\right] _{p,q}%
\sum_{k=0}^{n}p^{-[n^2+3n-k^2-k]/2}b_{n,k}^{p,q}(1,x)\qfrac{n}{k}_{p,q} \\
&&\qquad \times p^{[n^2+7n-k^2-5k]/2}\frac{[k+2]_{p,q}![n-k]_{p,q}!}{[n+3]_{p,q}!} \\
\!\!\!\! &=&\!\!\!\!\sum_{k=0}^{n}p^{2(n-k)}b_{n,k}^{p,q}(1,x)\frac{[k+1]_{p,q}[k+2]_{p,q}}{%
[n+2]_{p,q}[n+3]_{p,q}} \\
\!\!\!\! &=&\!\!\!\!\frac{1}{[n+2]_{p,q}[n+3]_{p,q}}%
\sum_{k=0}^{n}p^{2n-2k}b_{n,k}^{p,q}(1,x)(p^{k}+q[k]_{p,q}).
(p^{k+1}+qp^k+q^2[k]_{p,q}) \\
\!\!\!\! &=&\!\!\!\!\frac{1}{[n+2]_{p,q}[n+3]_{p,q}}\sum_{k=0}^{n}p^{2n-2k}b_{n,k}^{p,q}(1,x)
\\
&&\qquad\times [p^{2k}(p+q)+p^k[k]_{p,q} (2q^2+qp)+q^3[k]_{p,q}^2] \\
\!\!\!\! &=&\!\!\!\!\frac{p^{2n}[2]_{p,q}}{[n+2]_{p,q}[n+3]_{p,q}}
\sum_{k=0}^{n}b_{n,k}^{p,q}(1,x) \\
&&\qquad +\frac{(2q^2+qp)p^n[n]_{p,q}}{
[n+2]_{p,q}[n+3]_{p,q}}\sum_{k=0}^{n}b_{n,k}^{p,q}(1,x)\Bigl(p^{n-k}\frac{%
[k]_{p,q}}{[n]_{p,q}}\Bigr) \\
&&\qquad +\frac{q^3[n]_{p,q}^2}{[n+2]_{p,q}[n+3]_{p,q}}
\sum_{k=0}^{n}b_{n,k}^{p,q}(1,x)\left(p^{2n-2k}\frac{[k]_{p,q}^2}{[n]_{p,q}^2
}\right) \\
\!\!\!\! &=&\!\!\!\!\frac{p^{2n}[2]_{p,q}}{[n+2]_{p,q}[n+3]_{p,q}}B_{n,p,q}\left( 1;x\right) + 
\frac{(2q^2+qp)p^n[n]_{p,q}}{[n+2]_{p,q}[n+3]_{p,q}}B_{n,p,q}\left(
t;x\right) \\
&&\qquad +\frac{q^3[n]_{p,q}^2}{[n+2]_{p,q}[n+3]_{p,q}}B_{n,p,q}\left( t^2;x\right),
\end{eqnarray*}}
i.e.,
\begin{eqnarray*}
D_{n}^{p,q}(t^2;x)\!\!\!\! &=&\!\!\!\!\frac{p^{2n}[2]_{p,q}}{[n+2]_{p,q}[n+3]_{p,q}} +\frac{
(2q^2+qp)p^n[n]_{p,q}x}{[n+2]_{p,q}[n+3]_{p,q}}\\[2mm]
&& \qquad  +\frac{q^3[n]_{p,q}[x^2[n]_{p,q}+p^{n-1}x(1-x)]}{[n+2]_{p,q}[n+3]_{p,q}}. \quad\qed	
\end{eqnarray*}
%D_{n}^{p,q}(t^2;x)=\frac{p^{2n}[2]_{p,q}}{[n+2]_{p,q}[n+3]_{p,q}} +\frac{
%(2q^2+qp)p^n[n]_{p,q}x}{[n+2]_{p,q}[n+3]_{p,q}} +\frac{
%q^3[n]_{p,q}[x^2[n]_{p,q}+p^{n-1}x(1-x)]}{[n+2]_{p,q}[n+3]_{p,q}}. \]}
%\end{proof}

\begin{lemma}
\label{l2} Let $n > 3$ be a given natural number and let $0<q<p\leq 1$, $q_{0}
= q_{0}(n) \in (0,p)$ be the least number such that $$
p^{2n+1}q-p^{n+1}q^{n+1}+p^{2n-1}q^3-p^{n-1}q^{n+3}+p^{2n}q^2-p^nq^{n+2}-2p^{2n+3}+2p^nq^{n+3}>0
$$ for every $q \in (q_{0},1)$. Then
$$
D_{n}^{p,q}((t-x)^{2},x) \  \leq \  \frac{2}{[n+2]_{p,q}} \  \left(
\varphi^{2}(x) + \frac{1}{[n+3]_{p,q}} \right),
$$
\noindent where $\varphi^{2}(x) = x ( 1 - x )$, $x \in [0,1]$.
\end{lemma}

\begin{proof}
In view of Lemma \ref{l1}, we obtain
{\small\begin{eqnarray}
D_{n}^{p,q}((t-x)^{2},x)\!\!\!\! &=&\!\!\!\!x^{2} \, \frac{q^{3} [n]_{p,q} ([n]_{p,q} -
p^{n-1}) - 2 q [n]_{p,q} [n+3]_{p,q} + [n+2]_{p,q} [n+3]_{p,q}}{[n+2]_{p,q}
[n+3]_{p,q}}  \nonumber \\
& & + \ x \,  \frac{p^{n-1}q (p+q)^{2} [n]_{p,q} - 2 p^n[n+3]_{p,q}}{%
[n+2]_{p,q} [n+3]_{p,q}} + \frac{p^{2n}(p+q)}{[n+2]_{p,q} [n+3]_{p,q}}
\nonumber
\end{eqnarray}}

\noindent By direct computations, using the definition of the $(p,q)$%
-numbers, we get
{\small\begin{eqnarray*}
p^{n-1}q (p+q)^{2} [n]_{p,q} - 2 p^n[n+3]_{p,q} \!\!&=&\!\! p^{n-1}q (p + q)^{2}
\frac{p^n-q^n}{p-q} - 2p^n \frac{p^{n+3}-q^{n+3}}{p-q}   \\
\!\!&=&\!\!\frac{1}{p-q}[p^{2n+1}q-p^{n+1}q^{n+1}+p^{2n-1}q^3-p^{n-1}q^{n+3} 
\\[1mm]
&&\ \ +\ p^{2n}q^2-p^nq^{n+2}-2p^{2n+3}+2p^nq^{n+3}]> 0,   
\end{eqnarray*}}
for every $q \in (q_{0},1)$.
Furthermore,
%\begin{eqnarray}
$p^{n-1}q (p+q)^{2} [n]_{p,q} - 2 p^n[n+3]_{p,q}\leq  \ 2 [n+3]_{p,q}$
%\nonumber
%\end{eqnarray}
and following \cite{amc}, we have
\begin{eqnarray*}
&&p^{n-1}q (p+q)^{2} [n]_{p,q} - 2 p^n[n+3]_{p,q}+
q^{3} [n]_{p,q} ([n]_{p,q} - p^{n-1}) \\
&&\qquad\qquad\qquad\qquad - 2 q [n]_{p,q} [n+3]_{p,q} +
[n+2]_{p,q} [n+3]_{p,q}\\[1mm]
&&\qquad\quad= p^{n-1}q (p+q)^{2} [n]_{p,q} - 2p^n ( p^{n+2}+qp^{n+1}+q^2p^n+ q^{3}[n]_{p,q }) \\
&&\qquad\quad + q^{3} [n]_{p,q}^{2} - q^{3}p^{n-1} [n]_{p,q}- 2q [n]_{p,q} (p^{n+2}+qp^{n+1}+q^2p^n+ q^{3}[n]_{p,q}) \\
&&\qquad\quad+ ( p^{n+1}+qp^n+q^2[n]_{p,q}) ( p^{n+2}+qp^{n+1}+q^2p^n+ q^{3} [n]_{p,q
} )\le 0 .
\end{eqnarray*}

In conclusion, for $x \in [0,1],$ we have
{\small
\begin{eqnarray*}
&&D_{n}^{p,q}((t-x)^{2},x)=\frac{p^{2n}(p+q)}{[n+2]_{p,q} [n+3]_{p,q}}+\frac{p^{n-1}q (p+q)^{2} [n]_{p,q} - 2 p^n[n+3]_{p,q}}{[n+2]_{p,q}
[n+3]_{p,q}}\,  x ( 1 - x )  \\
&&\qquad\qquad\qquad 
+ \biggl( \frac{p^{n-1}q (p+q)^{2} [n]_{p,q} -
2 p^n[n+3]_{p,q}}{[n+2]_{p,q} [n+3]_{p,q}}   \\
&&\qquad\qquad\qquad  + \  \frac{q^{3} [n]_{p,q} ([n]_{p,q} - p^{n-1}) - 2 q [n]_{p,q}
[n+3]_{p,q} + [n+2]_{p,q} [n+3]_{p,q}}{[n+2]_{p,q} [n+3]_{p,q}} \biggr)
  x^{2}    \\
&&\qquad\qquad\qquad \leq  \frac{2 [n+3]_{p,q}}{[n+2]_{p,q} [n+3]_{p,q}}   \varphi^{2}(x) +
\frac{2}{[n+2]_{p,q} [n+3]_{p,q}}\\
&&\qquad\qquad\qquad\leq \frac{2}{[n+2]_{p,q}}   \left(
\varphi^{2}(x) + \frac{1}{[n+3]_{p,q}} \right),  
\end{eqnarray*}}
%
%
%\noindent$\displaystyle D_{n}^{p,q}((t-x)^{2},x) \ =$
%\begin{eqnarray}
%&=& \frac{p^{n-1}q (p+q)^{2} [n]_{p,q} - 2 p^n[n+3]_{p,q}}{[n+2]_{p,q}
%[n+3]_{p,q}} \cdot x ( 1 - x ) + \left( \frac{p^{n-1}q (p+q)^{2} [n]_{p,q} -
%2 p^n[n+3]_{p,q}}{[n+2]_{p,q} [n+3]_{p,q}}\right.  \nonumber \\
%& & \left. + \  \frac{q^{3} [n]_{p,q} ([n]_{p,q} - p^{n-1}) - 2 q [n]_{p,q}
%[n+3]_{p,q} + [n+2]_{p,q} [n+3]_{p,q}}{[n+2]_{p,q} [n+3]_{p,q}} \right)
%\cdot x^{2} + \frac{p^{2n}(p+q)}{[n+2]_{p,q} [n+3]_{p,q}}  \nonumber \\
%&\leq& \frac{2 [n+3]_{p,q}}{[n+2]_{p,q} [n+3]_{p,q}} \cdot \varphi^{2}(x) +
%\frac{2}{[n+2]_{p,q} [n+3]_{p,q}} \leq \frac{2}{[n+2]_{p,q}} \cdot \left(
%\varphi^{2}(x) + \frac{1}{[n+3]_{p,q}} \right),  \nonumber
%\end{eqnarray}
%
which was to be proved.
\end{proof}

\section{Direct Estimates}\label{sec:section4}
 \vglue-10pt
 \indent

We denote $W^{2}=\bigl\{g\in C[0,1]:g'',g''\in
C[0,1]\bigr\}$ for $\delta >0$, $K-$functional is defined as
$$
K_{2}(f,\delta )=\inf \bigl\{||f-g||+\eta ||g''||:g\in W^{2}\bigr\},
$$ 
where norm-$\|.\|$ denotes the uniform norm on $C[0,1]$. Following the well
known inequality due to DeVore and Lorentz \cite{Devore}, there exists a
absolute constant $C>0$ such that
\begin{equation}\label{3.3.9}
K_{2}(f,\delta )\leq C\omega _{2}(f,\sqrt{\delta }),  
\end{equation}%
where the second order modulus of smoothness for $f\in C[0,1]$ is defined as
$$
\omega _{2}(f,\sqrt{\delta })=\sup_{0<h\leq \sqrt{\delta }}\sup_{x,x+h\in
\lbrack 0,1]}|f(x+h)-f(x)|.
$$%
The usual modulus of continuity for $f\in C[0,1]$ is defined as
$$
\omega (f,\delta )=\sup_{0<h\leq \delta }\sup_{x,x+h\in \lbrack
0,1]}|f(x+h)-f(x)|.
$$
Our first main result is the following local theorem:
\begin{theorem}
\label{T-d1} Let $n > 3$ be a natural number and let $0<q<p\leq 1$, $q_{0} =
q_{0}(n) \in (0,p)$ be defined as in Lemma \ref{l2}. Then there exists an
absolute constant $C > 0$ such that
$$
\vert D_{n}^{p,q}(f,x) - f(x) \vert \,  \leq \, C  \omega_{2}\left(f ,
[n+2]_{p,q}^{-1/2} \delta_{n}(x)\right) + \omega \left(f,\frac{1-x}{%
[n+2]_{p,q}}\right),
$$
where $f \in C[0,1]$, $\DS\delta_{n}^{2}(x) = \varphi^{2}(x) + \dfrac{1}{[n+3]_{p,q}},$ $x \in [0,1]$ and $q \in (q_{0},1).$
\end{theorem}

\begin{proof}
For $f \in C[0,1]$ we define
$$
\widetilde{D}_{n}^{p,q}(f,x) \ = \ D_{n}^{p,q}(f,x) + f(x) - f\left( \frac{%
p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}} \right).
$$

\noindent Then, by Lemma \ref{l1}, we immediately get
\begin{equation}  \label{3.3.10}
\widetilde{D}_{n}^{p,q}(1,x) = D_{n}^{p,q}(1,x) = 1
\end{equation}
and
\begin{equation}  \label{3.3.11}
\widetilde{D}_{n}^{p,q}(t,x) = D_{n}^{p,q}(t,x) + x - \frac{p^{n}+q[n]_{p,q}x}{%
[n+2]_{p,q}} = x.
\end{equation}
By Taylor's formula
$$
g(t) = g(x) + (t-x) g'(x) + \int_{x}^{t} (t-u) g''(u)\,du,
$$

\noindent we get
\begin{eqnarray}
\widetilde{D}_{n}^{p,q}(g,x) &=& g(x) + \widetilde{D}_{n}^{p,q}\left( \int_{x}^{t} \
(t-u) \ g''(u) \ du , x \right)  \nonumber \\
&=& g(x) + D_{n}^{p,q}\left( \int_{x}^{t}(t-u)g''(u) \,
du , x \right)  \nonumber \\
&&- \int_{x}^{\frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}}\left( \frac{%
p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}} - u \right) g''(u) \, du.
\nonumber
\end{eqnarray}

\noindent Thus
\begin{eqnarray}  \label{3.3.12}
\displaystyle \vert \widetilde{D}_{n}^{p,q}(g,x) - g(x)
\vert \
&\leq& D_{n}^{p,q}\left( \Biggm \vert \int_{x}^{t} \  \vert t-u \vert 
\vert g''(u) \vert \, du \Biggm \vert , x \right)\nonumber\\&& 
+ \Biggm
\vert \int_{x}^{\frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}} \  \Biggm \vert \frac{%
p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}} - u \Biggm \vert  \vert g^{\prime
\prime }(u) \vert \ du \Biggm \vert  \nonumber \\
&\leq& D_{n}^{p,q}( (t-x)^{2} , x )  \Vert g''\Vert +
\left( \frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}} - x \right)^{2} \Vert
g''\Vert
\end{eqnarray}

\noindent Also, we have
\begin{eqnarray}\label{3.3.13}
&&D_{n}^{p,q}( (t-x)^{2} , x ) + \left( \frac{%
p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}} - x \right)^{2}\\[2mm]
&&\quad \leq \  \frac{2}{[n+2]_{p,q}} \left( \varphi^{2}(x) + \frac{1}{[n+3]_{p,q}}
\right) + \left( \frac{p^n - ( [n+2]_{p,q} - q [n]_{p,q} ) x}{[n+2]_{p,q}}
\right)^{2}.\nonumber	
\end{eqnarray}
Obviously
\begin{equation}\label{3.3.14}
1 \leq [n+2]_{p,q} - q [n]_{p,q} \leq 2.
\end{equation}

\noindent Then, using (\ref{3.3.14}), we get
\begin{eqnarray*}
&&\left( \frac{p^n - ( [n+2]_{p,q} - q [n]_{p,q} ) x}{ 
[n+2]_{p,q}} \right)^{2} \delta_{n}^{-2}(x)\\[2mm]
&&\qquad\quad =\frac{p^{2n} - 2 p^n( [n+2]_{p,q} - q [n]_{p,q} ) x 
+ ( [n+2]_{p,q} - q
[n]_{p,q} )^{2} x^{2}}{[n+2]_{p,q}^{2}}\\
&& \qquad\qquad\qquad \times\frac{[n]_{p,q}}{[n]_{p,q} x (
1 - x ) + 1} \\[2mm]
&&\qquad\quad\leq \frac{p^{2n} - 2 p^nx + 4 x^{2}}{[n+2]_{p,q}} \cdot \frac{[n]_{p,q}}{[n+2]_{p,q}} \cdot \frac{1}{[n]_{p,q} x ( 1 - x ) + 1},
\end{eqnarray*}
i.e.,
\begin{equation}\label{3.3.15}
\left( \frac{p^n - ( [n+2]_{p,q} - q [n]_{p,q} ) x}{ 
[n+2]_{p,q}} \right)^{2} \delta_{n}^{-2}(x)\leq \frac{3}{
[n+2]_{p,q}},	
\end{equation}
for $n \in \mathbb{N}$ and $0 < q < p\le 1.$ \ In conclusion, by 
(\ref{3.3.13}) and (\ref{3.3.15}), for $x \in [0,1]$, we obtain

\begin{equation}  \label{3.3.16}
D_{n}^{p,q}( (t-x)^{2} , x ) + \left( \frac{p^n + q [n]_{p,q} x}{[n+2]_{p,q}}
- x \right)^{2} \leq \frac{5}{[n+2]_{p,q}} \delta_{n}^{2}(x).
\end{equation}

\noindent Hence, by (\ref{3.3.12}) and with the conditions $n > 3$ and $x
\in [0,1],$ we have
\begin{equation}  \label{3.3.17}
\vert \widetilde{D}_{n}^{p,q}(g,x) - g(x) \vert \leq \frac{5}{[n+2]_{p,q}} \,%
\delta_{n}^{2}(x) \,\Vert g''\Vert.
\end{equation}
\noindent Furthermore, for $f\in C[0,1]$ we have $||D_{n}^{p,q}(f,x)||\le
||f||$, thus
\begin{equation}  \label{3.3.18}
\vert \widetilde{D}_{n}^{p,q}(f,x) \vert \leq \vert D_{n}^{p,q}(f,x) \vert +
\vert f(x) \vert + \Biggm \vert f\left( \frac{p^n + q [n]_{p,q} x}{%
[n+2]_{p,q}} \right) \Biggm \vert \leq 3 \Vert f \Vert.
\end{equation}
\noindent for all $f \in C[0,1].$

Now, for $f \in C[0,1]$ and $g \in W^{2},$ we obtain
\smallskip

\noindent$\displaystyle \vert D_{n}^{p,q}(f,x) - f(x) \vert$%
\begin{eqnarray*}
&&= \Biggm \vert \widetilde{D}_{n}^{p,q}(f,x) - f(x) + f\left( \frac{p^n + q
[n]_{p,q} x}{[n+2]_{p,q}} \right) - f(x) \Biggm \vert  \nonumber \\
&&\leq \vert \widetilde{D}_{n}^{p,q}(f-g,x) \vert + \vert \widetilde{D}
_{n}^{p,q}(g,x) - g(x) \vert + \vert g(x) - f(x) \vert\\[2mm]
&&\qquad\qquad\qquad\qquad + \Biggm \vert 
f\left( \frac{p^n + q [n]_{p,q} x}{[n+2]_{p,q}} \right) - f(x) \Biggm \vert
\nonumber \\
&&\leq 4 \  \Vert f - g \Vert + \frac{5}{[n+2]_{p,q}} \cdot \delta_{n}^{2}(x)
\cdot \Vert g''\Vert + \omega \Biggl( f , \Biggm \vert \frac{ 
p^n - ( [n+2]_{p,q} - q [n]_{p,q} ) x}{[n+2]_{p,q}} \Biggm \vert \Biggr)
\nonumber \\
&&\leq 5 \left( \Vert f - g \Vert + \frac{1}{[n+2]_{p,q}} \cdot
\delta_{n}^{2}(x) \cdot \Vert g''\Vert \right) + \omega
\left(f , \frac{1-x}{[n+2]_{p,q}} \right),  
\end{eqnarray*}
\noindent where we have used (\ref{3.3.17}) and (\ref{3.3.18}). Taking the
infimum on the right hand side over all $g \in W^{2},$ we obtain at once
$$
\vert D_{n}^{p,q}(f,x) - f(x) \vert \leq 5 \ K_{2}\left( f , \frac{1}{%
[n+2]_{p,q}} \delta_{n}^{2}(x) \right) + \omega \left( f , \frac{1-x}{%
[n+2]_{p,q}} \right).
$$
Finally, in view of (\ref{3.3.9}), we find
$$
|D_{n}^{p,q}(f,x)-f(x)|\  \leq \ C\  \omega _{2}\left(
f,[n+2]_{p,q}^{-1/2}\delta _{n}(x)\right) +\omega \left( f,\frac{1-x}{%
[n+2]_{p,q}}\right).
$$

\noindent This completes the proof of the theorem.
\end{proof}

The weighted modulus of continuity of second order for $f\in C[0,1]$ and $\varphi
(x)=\sqrt{x(1-x)}$ is defined as:
$$
\omega _{2}^{\varphi }(f,\sqrt{\delta })\ =\  \sup_{0<h\leq \sqrt{\delta } 
}\ \sup_{x,x\pm h\varphi \in \lbrack 0,1]}|f(x+h\varphi (x))-2f(x)+f(x-h\varphi
(x))|
$$%
\noindent the corresponding $K$ functional is defined by
$$
\overline{K}_{2,\varphi }(f,\delta )\ =\  \inf \{ \Vert f-g\Vert +\delta
\Vert \varphi ^{2}g''\Vert+\delta^2
\Vert g''\Vert:g\in W^{2}(\varphi ) \},
$$%
where
$$
W^{2}(\varphi )=\bigl\{g\in C[0,1]:g^{\prime }\in AC_{loc}[0,1],\varphi
^{2}g''\in C[0,1]\bigr\}
$$%
and $g'\in AC_{loc}[0,1]$ means that $g$ is differentiable and $%
g^{\prime }$ is absolutely continuous on every closed interval $[a,b]\subset
\lbrack 0,1].$ By the property due to Ditzian-Totik (see \cite[p. 24,
Theorem 1.3.1]{Ditzian}), we have
\begin{equation}
\overline{K}_{2,\varphi }(f,\delta )\  \leq \ C\  \omega _{2}^{\varphi }(f,%
\sqrt{\delta })  \label{3.3.19}
\end{equation}%
\noindent for some absolute constant $C>0.$ Moreover, with $\psi$ the admissible step-weight function on $[0,1],$ the Ditzian-Totik
moduli of first order is given by
$$
\vec{\omega}_{\psi }(f,\delta )\ =\  \sup_{0<h\leq \delta }\sup_{x,x\pm h\psi
(x)\in \lbrack 0,1]}|f(x+h\psi (x))-f(x)|.
$$


Now we state and prove the following global direct result:

\begin{theorem}
\label{td2}Let $n>3$ be a natural number and let $0<q<p\leq
1$, $q_{0}=q_{0}(n)\in (0,p$ be defined as in Lemma \ref{l2}. Then there exists
an absolute constant $C>0$ such that$0<q<p\leq 1$, $q_{0}=q_{0}(n)\in (0,p)$
$$
\Vert D_{n}^{p,q}f-f\Vert \  \leq \ C\  \omega _{2}^{\varphi
}(f,[n+2]_{q}^{-1/2})\ +\  \vec{\omega}_{\psi }(f,[n+2]_{q}^{-1}),
$$%
where $f\in C[0,1],$ $q\in (q_{0},1)$ and $\psi (x)=1-x,$ $x\in \lbrack
0,1]$.
\end{theorem}

\begin{proof}
Let us consider
$$
\widetilde{D}_{n}^{p,q}(f,x)\ =\ D_{n}^{p,q}(f,x)+f(x)-f\left( \frac{%
p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}\right) ,
$$

\noindent where $f\in C[0,1].$ By Taylor's formula with $g\in W^{2}(\varphi )$, we have
$$
g(t)=g(x)+(t-x)\ g^{\prime }(x)+\int_{x}^{t}\ (t-u)\ g''(u)du,
$$

\noindent with the applications of (\ref{3.3.10}) and (\ref{3.3.11}), we
obtain
\begin{eqnarray*}
\widetilde{D}_{n}^{p,q}(g,x)&=&g(x)+D_{n}^{p,q}\left( \int_{x}^{t} t-u) 
g''(u)\ du,x\right)\\ 
&&\qquad
-\int_{x}^{\frac{p^{n}+q[n]_{p,q}x}{%
[n+2]_{p,q}}}\  \left( \frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}-u\right)
g''(u)\ du.
\end{eqnarray*}
Thus we can write
\begin{eqnarray}
\displaystyle|\widetilde{D}_{n}^{p,q}(g,x)-g(x)|&\leq& \ D_{n}^{p,q}\left( \Biggm \vert \int_{x}^{t}\ |t-u|\cdot |g^{\prime
\prime }(u)|\ du\Biggm \vert,x\right) \nonumber\\&&+\Biggm \vert \int_{x}^{\frac{%
p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}}\  \Biggm \vert \frac{p^{n}+q[n]_{p,q}x}{%
[n+2]_{p,q}}-u\Biggm \vert \cdot \ |g''(u)|\ du\Biggm \vert.
\label{3.3.20}
\end{eqnarray}

\noindent Also, the function $\delta_{n}^{2}$ is concave on $[0,1],$ we
have for $u = t + \tau ( x - t ),$ $\tau \in [0,1],$ the following estimate
$$
\frac{\vert t - u \vert}{\delta_{n}^{2}(u)} = \frac{\tau \vert x - t \vert}{%
\delta_{n}^{2}(t + \tau ( x - t ))} \leq \frac{\tau \vert x - t \vert}{%
\delta_{n}^{2}(t) + \tau ( \delta_{n}^{2}(x) - \delta_{n}^{2}(t) )} \leq
\frac{\vert t - x \vert}{\delta_{n}^{2}(x)}.
$$

\noindent Hence, by (\ref{3.3.20}), we obtain
{\small\begin{eqnarray*}
&&\displaystyle|\widetilde{D}_{n}^{p,q}(g,x)-g(x)|   \\
&&\leq D_{n}^{p,q}\left( \Biggm \vert \int_{x}^{t}\  \frac{|t-u|}{\delta
_{n}^{2}(u)} du\Biggm \vert,x\right)   \Vert \delta _{n}^{2}g''\Vert +\Biggm \vert \int_{x}^{\frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}} 
}\frac{\Bigm \vert \frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}-u\Bigm \vert}{ 
\delta _{n}^{2}(u)}\ du\Biggm \vert   \Vert \delta _{n}^{2}g''\Vert 
  \\
&&\leq  \frac{1}{\delta _{n}^{2}(x)}  D_{n}^{p,q}((t-x)^{2},x)  \Vert
\delta _{n}^{2}g''\Vert +\frac{1}{\delta _{n}^{2}(x)} 
\left( \frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}-x\right)^{2}  \Vert \delta
_{n}^{2}g''\Vert.   
\end{eqnarray*}}

\noindent For $x\in \lbrack 0,1],$ in view of (\ref{3.3.16}) and
$$
\delta _{n}^{2}(x)\cdot |g''(x)|=|\varphi^2(x)g''(x)|+\frac{1 
}{[n+2]_{p,q}}\cdot |g''(x)| \le \|\varphi^2g''\| +\frac{1}{ 
[n+2]_{p,q}}  \Vert g''\Vert ,
$$

\noindent we get
\begin{equation}
|\widetilde{D}_{n}^{p,q}(g,x)-g(x)|\leq \frac{5}{[n+2]_{p,q}}\cdot \left( \Vert
\varphi ^{2}g''\Vert +\frac{1}{[n+2]_{p,q}}\cdot \Vert
g''\Vert \right) \label{3.3.21}
\end{equation}

\noindent Obviously using $[n]_{p,q}\leq [n+2]_{p,q}$, (\ref{3.3.18}) and (\ref%
{3.3.21}), we find for $f \in C[0,1],$ that
\begin{eqnarray*}
&&|D_{n}^{p,q}(f,x)-f(x)|\le |\widetilde{D}_{n}^{p,q}(f-g,x)|\\
%&\leq &|\widetilde{D}_{n}^{p,q}(f-g,x)|
&&\qquad\qquad +|\widetilde{D}_{n}^{p,q}(g,x)-g(x)|+|g(x)-f(x)|+\Biggm \vert f\Bigl( \frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}}\Bigr) -f(x)\Biggm \vert \qquad\ \ \qquad \\
\!\!\!\! &\leq&\!\!\!\!4 \Vert f-g\Vert +\frac{5}{[n+2]_{p,q}}  \Vert \varphi
^{2}g''\Vert +\frac{5}{[n+2]_{p,q}}  \Vert g''\Vert  +\Biggm \vert f\Bigl( \frac{p^{n}+q[n]_{p,q}x}{[n+2]_{p,q}} 
\Bigr) -f(x)\Biggm \vert.  
\end{eqnarray*} 

Finally, taking the infimum on the right hand side over all $g\in
W^{2}(\varphi ),$ we obtain
\begin{eqnarray}\label{3.3.22}
|D_{n}^{p,q}(f,x)-f(x)|&\leq & 5\overline{K}_{2,\varphi }\left( f,\frac{1}{%
[n+2]_{p,q}}\right)\\
&& +\Biggm \vert f\left( \frac{p^{n}+q[n]_{p,q}x}{%
[n+2]_{p,q}}\right) -f(x)\Biggm \vert\nonumber  
\end{eqnarray}

Also, we have

\noindent $\displaystyle \Biggm \vert f\Bigl( \frac{p^{n}+q[n]_{p,q}x}{%
[n+2]_{p,q}}\Bigr) -f(x)\Biggm \vert  =\Biggm \vert f\Bigl( x+\psi (x)  \frac{p^{n}-([n+2]_{p,q}-q[n]_{p,q})x%
}{[n+2]_{p,q}\psi (x)}\Bigr) -f(x)\Biggm \vert$
\begin{eqnarray*}
%&=&\Biggm \vert f\left( x+\psi (x)\cdot \frac{p^{n}-([n+2]_{p,q}-q[n]_{p,q})x%
%}{[n+2]_{p,q}\psi (x)}\right) -f(x)\Biggm \vert \\
\!\!\!&\leq &\!\!\!\sup_{t,t+\psi (t) 
\frac{p^{n}-([n+2]_{p,q}-q[n]_{p,q})x}{[n+2]_{p,q}}\in \lbrack 0,1]}\Biggm \vert 
f\Bigl( t+\psi (t)\frac{p^{n}-([n+2]_{p,q}-q[n]_{p,q})x}{ 
[n+2]_{p,q}\psi (x)}\Bigr) -f(t)\Biggm \vert  \\[2mm]
&\leq &\vec{\omega}_{\psi }\left( f,\frac{|p^{n}-([n+2]_{p,q}-q[n]_{p,q})x|}{%
[n+2]_{p,q}\psi (x)}\right) \\[2mm]
&\leq & \vec{\omega}_{\psi }\left( f,\frac{1-x}{%
[n+2]_{p,q}\psi (x)}\right) =\vec{\omega}_{\psi }\left( f,\frac{1}{%
[n+2]_{p,q}}\right) . 
\end{eqnarray*}

\noindent Hence, by (\ref{3.3.22}) and (\ref{3.3.19}), we get
$$
\Vert D_{n}^{p,q}f-f\Vert \  \leq \ C\  \omega _{2}^{\varphi
}(f,[n+2]_{p,q}^{-1/2})\ +\  \omega _{\psi }(f,[n+2]_{p,q}^{-1}).
$$%
 This completes the proof of the theorem.
\end{proof}

\begin{remark}
For $q\in (0,1)$ and $p\in (q,1]$ it is obvious that $\lim_{n\rightarrow
\infty }\left[ n\right] _{p,q}={1}/{(p-q)}$. Thus above theorems do not
give a approximation result. If we choose $q_{n}=e^{-{1}/{n}}$ and $
p_{n}=e^{-{1}/{(n+1)}}$ such that $0<q_{n}<p_{n}\leq 1$,
$\lim\limits_{n\rightarrow \infty }p_{n}=\lim\limits_{n\rightarrow \infty }q_{n}=1$ and $
\lim\limits_{n\rightarrow \infty }p_{n}^{n}=\lim\limits_{n\rightarrow \infty
}q_{n}^{n}=1/e.$ Also we have
$$
\lim_{n\rightarrow \infty }\left[ n\right] _{p_{n},q_{n}}=\infty .
$$
Since $\left[ n+2\right] _{p,q}=\left[ 2\right] _{p,q}p^{n}+q^{2}\left[ n%
\right] _{p,q}$ we can write 
$$\lim_{n\rightarrow \infty }\frac{1}{\left[ n%
\right] _{p_{n},q_{n}}}=\lim_{n\rightarrow \infty }\frac{1}{\left[ n+2\right]
_{p_{n},q_{n}}}=0\quad\mbox{ and }\quad
\lim_{n\rightarrow \infty }\frac{\left[ n\right]
_{p_{n},q_{n}}}{\left[ n+2\right] _{p_{n},q_{n}}}=1.$$
\end{remark}
\begin{figure} [h]
\centering
\includegraphics[width=0.5\textwidth]{figuref1.eps}% 
\includegraphics[width=0.5\textwidth]{figuref2.eps}%
\caption{\rm Functions $f_1(x)=0.5e^x$ and $f_2(x)=\ln(1+x)$} \label{figuref1}
\end{figure}

\begin{example}
With the help of {\sc Mathematica}, we show comparisons and some illustrative
graphics for the convergence of $(p,q)$-Bernstein-Durrmeyer
operators $D_{n}^{p,q}(f;x)$ for different values of the parameters $p $ and $q.$
We have considered different functions as
shown in Figures \ref{figuref1}.
\end{example}


%\begin{figure} [htb]
%\centering
%\includegraphics[height=4.5cm]{figuref2.eps}
%\caption{\rm Function $f_2(x)=\ln(1+x)$} \label{figuref2}
%\end{figure}
\bigskip

\begin{thebibliography}{999999}
\bibitem{avr}{\sc A. \ Aral, V. \ Gupta {\rm and} R.\ P.\ Agarwal}:\textit{ Applications of $q$
Calculus in Operator Theory}, Springer 2013.

\bibitem{Devore} {\sc R.\ A.\  DeVore {\rm and} G.\  G.\ Lorentz}:\textit{ Constructive Approximation},
Springer, Berlin 1993.

\bibitem{Ditzian} {\sc Z.\ Ditzian {\rm and} V.\ Totik}: \textit{Moduli of Smoothness}, Springer ,
New York 1987.

\bibitem{ZFVG} {\sc Z.\ Finta {\rm and} V.\ Gupta}: \textit{Approximation by $q$-Durrmeyer
operators}, J. Appl. Math. Computing 29 (1-2)(2009), 401-415.

\bibitem{amc} {\sc V.\ Gupta}: \textit{Some approximation properties on $q$-Durrmeyer
operators}, Appl. Math. Computation 197 (1) (2008), 172-178.

\bibitem{vgrpa} {\sc V.\ Gupta {\rm and} R.\ P.\ Agarwal}:\textit{ Convergence Estimates in
Approximation Theory}, Springer, 2014.

\bibitem{1} {\sc K.\ Khan {\rm and} D.\ K.\ Lobiyal}:\textit{ B\'ezier curves based on Lupas $(p,q)$%
-analogue of Bernstein polynomials in CAGD}, arXiv:1505.01810 [cs.GR].

\bibitem{4} {\sc M.\ Mursaleen, K.\ J.\ Ansari {\rm and} A. Khan}:\textit{ On $(p,q)$-analogue of
Bernstein operators}, Appl. Math. Comput. 266 (2015) 874-882.

\bibitem{2} {\sc P.\ N.\ Sadjang}: \textit{On the $(p,q)$-Gamma and the $(p,q)$-Beta
functions}, arXiv 1506.07394v1. 22 Jun 2015.

\bibitem{3} {\sc P.\ N.\ Sadjang}: \textit{On the fundamental theorem of $(p,q)$ -calculus
and some $(p,q)$-Taylor formulas}, arXiv:1309.3934 [math.QA].
\bibitem{VSSY} {\sc V.\ Sahai {\rm and} S.\ Yadav}: \textit{Representations of two parameter quantum algebras and $p,q$-special functions}, J. Math. Anal. Appl. 335(2007), 268-279.




\end{thebibliography}

 \bigskip

{\small\rm\baselineskip=10pt
 \baselineskip=10pt
 \qquad Vijay Gupta\par
 \qquad Department of Mathematics\par
 \qquad Netaji Subhas Institute of Technology\par
 \qquad Sector 3 Dwarka\par
 \qquad New Delhi 110078, India\par
 \qquad {\tt vijaygupta2001@hotmail.com}

 \bigskip \smallskip

 \qquad Ali Aral\par
 \qquad Department of Mathematics\par
 \qquad Kirikkale University\par
 \qquad TR-71450 Yahsihan Kirikkale/ Turkey\par
 \qquad {\tt aliaral73@yahoo.com}
 }

 \end{document}
