1. Introduction
Since Ostrowski first time proved his inequality in 1938, after that many
researchers did a lot of work on it. Some monographs presented by Barnet
et al., [1] and Dragomir et al. [2] on
Ostrowski’s type inequalities. In the past years, many researchers [3,4,5,6,7] did efforts to obtain tighter error bounds of Ostrowski type
inequalities. Inspired and motivated by the work of above
famous Mathematician [8,9] and [2,10,11], we started our work to extend and produce new and generalized
Ostrowski’s integral inequalities.
In this paper, we introduced some new generalized
different types of Kernels, development of new identities and new error
bounds of Ostrowski’s type inequalities for first and second derivable
mappings. By utilizing our obtained results, previous famous results are
recaptured as special cases.
2. Results for Quadratic mapping
Theorem 1.
Let \(J\subseteq \mathbb{R} \) such that \(c,d\in J, \) and \(c< d. \) If \(s: J\rightarrow\mathbb{R} \) is a derivable function such that \(\gamma \leq s^{\prime
}\left( t\right) \leq \Gamma , \) and \(\varphi ,\psi ,\gamma ,\Gamma\in\mathbb{R}, \) then we get
\begin{align}
\label{3.27}
& \left\vert \frac{1}{\varphi +\psi }\left[ \frac{1}{2}\left( \varphi \left(
x-c\right) ^{2}-\psi \left( x-d\right) ^{2}\right) s^{\prime }\left(
x\right) \right. \right. \left. -\left( \varphi \left( x-c\right) +\psi \left( x-d\right) \right)
s\left( x\right) +\left( \varphi \int\limits_{c}^{x}s\left( t\right)
dt+\psi \int\limits_{x}^{d}s\left( t\right) dt\right) \right. \notag \\
& \left. \left. -\frac{\Gamma +\gamma }{12}\left( \varphi \left( x-c\right)
^{3}-\psi \left( x-d\right) ^{3}\right) \right] \right\vert \leq \frac{1}{\varphi +\psi }\frac{\Gamma -\gamma }{12}\left( \varphi
\left( x-c\right) ^{3}-\psi \left( x-d\right) ^{3}\right),
\end{align}
(1)
for all \(t\in \left[ c,d\right]. \)
Proof.
Define a new peano Kernel \(L\left( x,t\right) :\left[ c,d\right]
\rightarrow\mathbb{R} \) by
\begin{equation}
L\left( x,t\right) =\left\{
\begin{array}{ccc}
\frac{\varphi }{\varphi +\psi }\frac{\left( t-c\right) ^{2}}{2}, & & \ \
t\in \left[ c,x\right] \\
\frac{\psi }{\varphi +\psi }\frac{\left( t-d\right) ^{2}}{2}, & & \ \ t\in
(x,d]
\end{array}
\right. \label{M-1}
\end{equation}
(2)
for all \( x\in \left[ c,d\right]. \) By using (2), we get
\begin{align}
\label{3.28}
\int\limits_{c}^{d}L\left( x,t\right) s^{\prime \prime }\left( t\right) dt
& =\frac{1}{\varphi +\psi }\left[ \frac{\varphi }{2}\left( x-c\right)
^{2}s^{\prime }\left( x\right) -\frac{\psi }{2}\left( x-d\right)
^{2}s^{\prime }\left( x\right) -\varphi \left( x-c\right) \right. \notag \\
&\;\;\; \left. \times s\left( x\right) +\psi \left( x-d\right) s\left( x\right)
+\left( \varphi \int\limits_{c}^{x}s\left( t\right) dt+\psi
\int\limits_{x}^{d}s\left( t\right) dt\right) \right] .
\end{align}
(3)
Again, by using (2), we get
\begin{align}
\label{3.29}
\int\limits_{c}^{d}L\left( x,t\right) dt
& =\frac{1}{6\left( \varphi +\psi \right) }\left( \varphi \left( x-c\right)
^{3}-\psi \left( x-d\right) ^{3}\right) .
\end{align}
(4)
Using (3) and (4), we get
\begin{align}
\label{3.30}
\int\limits_{c}^{d}L\left( x,t\right) \left( s^{\prime \prime }\left(
t\right) -C\right) dt
& =\frac{1}{\varphi +\psi }\left[ \frac{\varphi }{2}\left( x-c\right)
^{2}s^{\prime }\left( x\right) -\frac{\psi }{2}\left( x-d\right)
^{2}s^{\prime }\left( x\right) -\varphi \left( x-c\right) \right. s\left( x\right) +\psi \left( x-d\right) s\left( x\right) \notag \\
& \;\;\;\left.
+\left( \varphi \int\limits_{c}^{x}s\left( t\right) dt+\psi
\int\limits_{x}^{d}s\left( t\right) dt\right) \right.-\left. \frac{C}{6}\left( \varphi \left( x-c\right) ^{3}-\psi \left(
x-d\right) ^{3}\right) \right] .
\end{align}
(5)
On the other hand
\begin{equation}
\left\vert \int\limits_{c}^{d}L\left( x,t\right) \left( s^{\prime \prime
}\left( t\right) -C\right) \right\vert \leq \underset{t\in \left[ c,d\right]
}{\max }\ \left\vert s^{\prime \prime }\left( t\right) -C\right\vert
\int\limits_{c}^{d}L\left( x,t\right) dt. \label{3.31}
\end{equation}
(6)
\begin{equation}
\int\limits_{c}^{d}\left\vert L\left( x,t\right) \right\vert dt=\frac{1}{
6\left( \varphi +\psi \right) }\left( \varphi \left( x-c\right) ^{3}-\psi
\left( x-d\right) ^{3}\right) . \label{3.32}
\end{equation}
(7)
Let
\(C=\frac{\Gamma +\gamma }{2}
\),
then,
\(
\underset{t\in \left[ c,d\right] }{\max }\left\vert s^{\prime \prime }\left(
t\right) -C\right\vert \leq \frac{\Gamma -\gamma }{2}.
\)
Thus (6) becomes
\begin{align}
\label{3.35}
\left\vert \int\limits_{c}^{d}L\left( x,t\right) \left( s^{\prime \prime
}\left( t\right) -C\right) dt\right\vert
\leq \frac{\Gamma -\gamma }{2}\left[ \frac{1}{6\left( \varphi +\psi
\right) }\left( \varphi \left( x-c\right) ^{3}-\psi \left( x-d\right)
^{3}\right) \right] .
\end{align}
(8)
Using (5) in (8), we get our required result (1).
Remark 1.
By putting \(\varphi =\psi \) in (1), we get
\begin{align}
\label{3.37}
& \left\vert \left( d-c\right) \left( x-\frac{c+d}{2}\right) s^{\prime
}\left( x\right) -\left( d-c\right) s\left( x\right)
+\int\limits_{c}^{d}s\left( t\right) dt\right. \left. -\frac{\Gamma +\gamma }{2}\left( d-c\right) \left( \frac{\left(
d-c\right) ^{2}}{24}+\frac{1}{2}\left( x-\frac{c+d}{2}\right) ^{2}\right)
\right\vert \notag \\
& \leq \frac{\Gamma -\gamma }{2}\left( d-c\right) \left( \frac{\left(
d-c\right) ^{2}}{24}+\frac{1}{2}\left( x-\frac{c+d}{2}\right) ^{2}\right) .
\end{align}
(9)
Corollary 2.
By putting \(x=\frac{c+d}{2} \) in (9), we get mid point inequality:
\begin{align*}
& \left\vert \int\limits_{c}^{d}s\left( t\right) dt-\left( d-c\right)
s\left( \frac{c+d}{2}\right) -\frac{1}{48}\left( \Gamma +\gamma \right)
\left( d-c\right) ^{3}\right\vert \leq \frac{1}{48}\left( \Gamma -\gamma \right) \left( d-c\right) ^{3}.
\end{align*}
3. Applications in numerical integration
Using [
5], we suppose that \(J_{n}:c=x_{0}< x_{1}< x_{2} < ….< x_{n-1}< x_{n}=d \) a partition of \([c,d] \), \(\xi_{i}\in\left[ x_{i}+\delta\frac{\varrho_{i}}{2},x_{i+1}-\delta\frac{\varrho_{i}}{2}\right] , \) \(\left( i=0,1,…..,n-1\right) \) and \(\varrho_{i}=x_{i+1}-x_{i} \) \(,\left( i=0,1,…..,n-1\right), \) then following
theorem exist:
Theorem 3.
Let \(s:\left[ c,d\right] \rightarrow\mathbb{R} \) be continuous on \(\left[ c,d\right] \) and derivable on \(\left(c,d\right) , \) then following formula exist:
\begin{equation}
\int\limits_{c}^{d}s\left( t\right) dt=A\left( s,\xi ,J_{n}\right) +R\left(
s,\xi ,J_{n}\right), \label{3.41}
\end{equation}
(10)
where
\begin{align}
\label{3.42}
A\left( s,\xi ,J_{n}\right)\leq& \sum\limits_{i=0}^{n-1}\frac{1}{2\left( \varphi +\psi \right) }
\left( \varphi \left( \xi _{i}-x_{i}\right) ^{2}-\psi \left( \xi
_{i}-x_{i+1}\right) ^{2}\right) s^{\prime }\left( \xi _{i}\right) \notag \\
& -\sum\limits_{i=0}^{n-1}\frac{1}{\varphi +\psi }\left( \varphi \left( \xi
_{i}-x_{i}\right) +\psi \left( \xi _{i}-x_{i+1}\right) \right) s\left( \xi
_{i}\right),
\end{align}
(11)
\begin{equation}
R\left( s,\xi ,J_{n}\right) \leq \Gamma \sum\limits_{i=0}^{n-1}\frac{
\varrho _{i}}{6\left( \varphi +\psi \right) }\left( \varphi \left( \xi
_{i}-x_{i}\right) ^{3}-\psi \left( \xi _{i}-x_{i+1}\right) ^{3}\right) .
\label{3.43}
\end{equation}
(12)
and remainder satisfies the estimation for all \(\xi _{i}\in \left[
x_{i},x_{i+1}\right] . \)
Proof.
By using Theorem 1 on \(\left[ x_{i},x_{i+1}\right] ,\xi _{i}\in \left[ x_{i},x_{i+1}\right] , \) to get:
\begin{align}
\label{3.44}
& \left\vert \frac{1}{2\left( \varphi +\psi \right) }\left( \varphi \left(
\xi _{i}-x_{i}\right) ^{2}-\psi \left( \xi _{i}-x_{i+1}\right) ^{2}\right)
s^{\prime }\left( \xi _{i}\right) \right.
\left. -\frac{1}{\varphi +\psi }\left[ \varphi \left( \xi
_{i}-x_{i}\right) +\psi \left( \xi _{i}-x_{i+1}\right) \right] s\left( \xi
_{i}\right) \right. \notag \\
& \left. +\frac{1}{\varphi +\psi }\left( \varphi \int\limits_{x_{i}}^{\xi
_{i}}s\left( t\right) dt+\psi \int\limits_{\xi _{i}}^{x_{i+1}}s\left(
t\right) dt\right) \right.
\left. -\frac{\varrho _{i}}{12\left( \varphi +\psi \right) }\left( \Gamma
+\gamma \right) \left( \varphi \left( \xi _{i}-x_{i}\right) ^{3}-\psi \left(
\xi _{i}-x_{i+1}\right) ^{3}\right) \right\vert \notag \\
& \leq \frac{\varrho _{i}}{12\left( \varphi +\psi \right) }\left( \Gamma
-\gamma \right) \left( \varphi \left( \xi _{i}-x_{i}\right) ^{3}-\psi \left(
\xi _{i}-x_{i+1}\right) ^{3}\right),
\end{align}
(13)
or
\begin{align*}
& \left\vert \sum\limits_{i=0}^{n-1}\frac{1}{2\left( \varphi +\psi \right) }
\left( \varphi \left( \xi _{i}-x_{i}\right) ^{2}-\psi \left( \xi
_{i}-x_{i+1}\right) ^{2}\right) s^{\prime }\left( \xi _{i}\right) \right.
\left. -\sum\limits_{i=0}^{n-1}\frac{1}{\varphi +\psi }\left( \varphi
\left( \xi _{i}-x_{i}\right) +\psi \left( \xi _{i}-x_{i+1}\right) \right)
s\left( \xi _{i}\right) \right. \\
& \left. +\sum\limits_{i=0}^{n-1}\frac{1}{\varphi +\psi }\left( \varphi
\int\limits_{c}^{x}s\left( t\right) dt+\psi \int\limits_{x}^{d}s\left(
t\right) dt\right) \right.
\left. -\frac{\Gamma +\gamma }{2}\sum\limits_{i=0}^{n-1}\frac{\varrho _{i}
}{6\left( \varphi +\psi \right) }\left( \varphi \left( \xi _{i}-x_{i}\right)
^{3}-\psi \left( \xi _{i}-x_{i+1}\right) ^{3}\right) \right\vert \\
& \leq \frac{\Gamma -\gamma }{2}\sum\limits_{i=0}^{n-1}\frac{\varrho _{i}}{
6\left( \varphi +\psi \right) }\left( \varphi \left( \xi _{i}-x_{i}\right)
^{3}-\psi \left( \xi _{i}-x_{i+1}\right) ^{3}\right) .
\end{align*}
With the help of generalized triangular inequality, we get the
desired estimation.
4. Results for generalized linear mapping
Theorem 4.
Let \(r:I\rightarrow\mathbb{R} \) and \(v,w\in I, \) \(v< w. \) If \(g^{\prime }:I\rightarrow\mathbb{R} \), such that \(\gamma \leq r^{\prime }\left( t\right) \leq \Gamma ,\
\forall \) \(t\in \left[ v,w\right] \) and \(\varphi ,\psi ,\ \gamma ,\Gamma
\in\mathbb{R}. \) We have
\begin{align}
\label{a}
& \left\vert \frac{1}{\varphi +\psi }\left[ \varphi \left( x-v-\varrho \frac{
w-v}{2}\right) -\psi \left( x-w+\varrho \frac{w-v}{2}\right) \right] r\left(
x\right) \right. +\frac{\varrho \left( w-v\right) }{2\left( \varphi +\psi \right) }
\left( \varphi r\left( v\right) +\psi r\left( w\right) \right)\notag\\
& \left. -\frac{1}{
\varphi +\psi }\left( \varphi \int\limits_{v}^{x}r\left( t\right) wt+\psi
\int\limits_{x}^{w}r\left( t\right) wt\right) \right. \left. -\frac{\Gamma +\gamma }{4\left( \varphi +\psi \right) }\left(
\varphi \left( x-v-\varrho \frac{w-v}{2}\right) ^{2}-\psi \left( x-w+\varrho
\frac{w-v}{2}\right) ^{2}\right) \right. \notag \\
& \left. +\frac{\varrho ^{2}\left( \varphi -\psi \right) }{16\left( \varphi
+\psi \right) }\left( \Gamma +\gamma \right) \left( w-v\right)
^{2}\right\vert \notag \\
& \leq \frac{\Gamma -\gamma }{2}\left[ \frac{\varrho ^{2}}{8}\left(
w-v\right) ^{2}\right. \left. +\frac{1}{2\left( \varphi +\psi \right) }\left( \varphi \left(
x-v-\varrho \frac{w-v}{2}\right) ^{2}+\psi \left( x-w+\varrho \frac{w-v}{2}
\right) ^{2}\right) \right] .
\end{align}
(14)
Proof.
First we define the mapping \(L\left( x,t\right) \ :\left[ v,w\right]
\rightarrow\mathbb{R}
\) by
\begin{equation}
L\left( x,t\right) =\left\{
\begin{array}{ccc}
\frac{\varphi }{\varphi +\psi }\left[ t-\left( v+\varrho \frac{w-v}{2}
\right) \right] , & & \ \ t\in \left[ v,x\right] \\
\frac{\psi }{\varphi +\psi }\left[ t-\left( w-\varrho \frac{w-v}{2}\right)
\right] , & & \ \ t\in (x,w]
\end{array}
\right. \label{M-2}
\end{equation}
(15)
By using (15), we get
\begin{align}
\label{1}
\int\limits_{v}^{w}L\left( x,t\right) r^{\prime }\left( t\right) dt
=&\frac{1}{\varphi +\psi }\left[ \varphi \left( x-\left( v+\varrho \frac{w-v
}{2}\right) \right) r\left( x\right) -\psi \left( x-\left( w-\varrho \frac{
w-v}{2}\right) \right) \right. \notag \\
& \times r\left( x\right) +\left. \varrho \frac{w-v}{2}\left( \varphi
r\left( v\right) +\psi r\left( w\right) \right) -\left( \varphi
\int\limits_{v}^{x}r\left( t\right) dt+\psi \int\limits_{x}^{w}r\left(
t\right) dt\right) \right],
\end{align}
(16)
and
\begin{align}
\label{2}
\int\limits_{v}^{w}L\left( x,t\right) dt =&\frac{\varphi }{2\left( \varphi +\psi \right) }\left[ \left( x-\left(
v+\varrho \frac{w-v}{2}\right) \right) ^{2}-\frac{\varrho ^{2}}{4}\left(
w-v\right) ^{2}\right] \notag \\
& +\frac{\psi }{2\left( \varphi +\psi \right) }\left[ \frac{\varrho ^{2}}{4}
\left( w-v\right) ^{2}-\left( x-\left( w-\varrho \frac{w-v}{2}\right)
\right) ^{2}\right] .
\end{align}
(17)
We put \(C=\frac{\Gamma +\gamma }{2} \) and using (16) and (17), we get
\begin{align}
\label{3}
\int\limits_{v}^{w}L\left( x,t\right) \left( r^{\prime }\left( t\right)
-C\right) dt =&\frac{\varphi }{\varphi +\psi }\left[ \left( x-\left( v+\varrho \frac{w-v
}{2}\right) \right) r\left( x\right) +\frac{\varrho }{2}\left( w-v\right)
r\left( v\right) -\int\limits_{v}^{x}r\left( t\right) dt\right] \notag \\
& +\frac{\psi }{\varphi +\psi }\left[ \frac{\varrho }{2}\left( w-v\right)
r\left( w\right) -\left( x-\left( w-\varrho \frac{w-v}{2}\right) \right)
r\left( x\right) -\int\limits_{x}^{w}r\left( t\right) dt\right] \notag \\
& -\frac{C\varphi }{2\left( \varphi +\psi \right) }\left[ \left( x-\left(
v+\varrho \frac{w-v}{2}\right) \right) ^{2}-\frac{\varrho ^{2}}{4}\left(
w-v\right) ^{2}\right] \notag \\
& -\frac{C\psi }{2\left( \varphi +\psi \right) }\left[ \frac{\varrho ^{2}}{4}
\left( w-v\right) ^{2}-\left( x-\left( w-\varrho \frac{w-v}{2}\right)
\right) ^{2}\right] .
\end{align}
(18)
Let
\begin{equation*}
C=\frac{\Gamma +\gamma }{2}.
\end{equation*}
Then
\begin{equation}
\left\vert \int\limits_{v}^{w}L\left( x,t\right) \left( r^{\prime }\left(
t\right) -C\right) dt\right\vert \leq \underset{t\in \left[ v,w\right] }{
\max }\left\vert r^{\prime }\left( t\right) -C\right\vert
\int\limits_{v}^{w}\left\vert L\left( x,t\right) \right\vert dt. \label{4}
\end{equation}
(19)
Now
\begin{align}
\label{5}
\int\limits_{v}^{w}\left\vert L\left( x,t\right) \right\vert dt
=\frac{\varrho ^{2}}{8}\left( w-v\right) ^{2}+\frac{1}{2\left( \varphi
+\psi \right) } \left[ \varphi \left( x-v-\varrho \frac{w-v}{2}\right) ^{2}+\psi
\left( x-w+\varrho \frac{w-v}{2}\right) ^{2}\right] ,
\end{align}
(20)
\begin{equation}
\underset{t\in \left[ v,w\right] }{\max }\left\vert r^{\prime }\left(
t\right) -C\right\vert \leq \frac{\Gamma -\gamma }{2}\ \text{ for all }
\ \gamma \leq t\leq \Gamma . \label{6}
\end{equation}
(21)
Using (19) and (21), we have
\begin{align}
\label{7}
&\left\vert \int\limits_{v}^{w}L\left( x,t\right) \left( r^{\prime }\left(
t\right) -\frac{\Gamma +\gamma }{2}\right) dt\right\vert\notag\\ &\leq \frac{\Gamma -\gamma }{2}\left[ \frac{\varrho ^{2}}{8}\left(
w-v\right) ^{2}+\frac{1}{2\left( \varphi +\psi \right) }\right.
\left. \left( \varphi \left( x-v-\varrho \frac{w-v}{2}\right)
^{2}+\psi \left( x-w+\varrho \frac{w-v}{2}\right) ^{2}\right) \right].
\end{align}
(22)
Using (18) and (22), we get our required result (14).
Remark 2.
By putting \(\varrho =0 \) in (14), we get
\begin{align*}
& \left\vert \frac{1}{\varphi +\psi }\left[ \left( \varphi \left( x-v\right)
-\psi \left( x-w\right) \right) r\left( x\right) -\left( \varphi
\int\limits_{v}^{x}r\left( t\right) dt+\psi \int\limits_{x}^{w}r\left(
t\right) dt\right) \right. \right. \\
& \left. \left. -\frac{1}{4}\left( \Gamma +\gamma \right) \left( \varphi
\left( x-v\right) ^{2}-\psi \left( x-w\right) ^{2}\right) \right] \right\vert
\\
& \leq \frac{\Gamma -\gamma }{2}\left[ \frac{\varrho ^{2}}{8}\left(
w-v\right) ^{2}+\frac{1}{2\left( \varphi +\psi \right) }\left( \varphi
\left( x-v\right) ^{2}+\psi \left( x-w\right) ^{2}\right) \right] .
\end{align*}
5. Results for generalized Quadratic mapping
Theorem 5.
Let \(z:I\subseteq
\mathbb{R}, \) and \( c,d\in I,\ c< d. \) If \(z:I\rightarrow
\mathbb{R}
\) is a derivable function such that \(\gamma \leq z^{\prime
}\left( t\right) \leq \Gamma ,\ \forall \ t\in \left[ c,d\right] , \) the
constants \(\varphi ,\psi ,\ \gamma ,\Gamma \in
\mathbb{R}
. \) Then, we get
\begin{align}
\label{3.45}
& \left\vert \frac{1}{2\left( \varphi +\psi \right) }\left( \varphi \left(
x-c-\varrho \frac{d-c}{2}\right) ^{2}-\psi \left( x-d+\varrho \frac{d-c}{2}
\right) ^{2}\right) z^{\prime }\left( x\right) \right.
+\left. \frac{\varrho ^{2}}{8\left( \varphi +\psi \right) }\left(
d-c\right) ^{2}\left( \psi z^{\text{ }\prime }\left( d\right) -\varphi
z^{\prime }\left( c\right) \right) \right. \notag \\
& +\left. \frac{1}{\varphi +\psi }\left( \psi \left( x-d+\varrho \frac{d-c}{2
}\right) -\varphi \left( x-c-\varrho \frac{d-c}{2}\right) \right) z\left(
x\right) \right. -\left. \frac{\varrho }{2\left( \varphi +\psi \right) }\left( d-c\right)
\left( \varphi z\left( c\right) +\psi z\left( d\right) \right) \right.
\notag \\
& \left. +\frac{1}{\varphi +\psi }\left( \varphi \int\limits_{c}^{x}z\left(
t\right) dt+\psi \int\limits_{x}^{d}z\left( t\right) dt\right) -\frac{
\Gamma +\gamma }{2}\left[ \frac{\varrho ^{3}}{48}\left( d-c\right)
^{3}\right. \right. \left. +\frac{1}{6\left( \varphi +\psi \right) }\left( \varphi
\left( x-c-\varrho \frac{d-c}{2}\right) ^{3}\right. \right.\notag \\
& \left.\left.\left. -\psi \left( x-d+\varrho \frac{
d-c}{2}\right) ^{3}\right) \right] \right\vert \notag \\
& \leq \frac{\Gamma -\gamma }{2}\left[ \frac{\varrho ^{3}\left( \psi
-\varphi \right) }{48\left( \varphi +\psi \right) }\left( d-c\right) ^{3}\
\right.
\left. +\frac{1}{6\left( \varphi +\psi \right) }\left( \varphi \left(
x-d+\varrho \frac{d-c}{2}\right) ^{3}+\psi \left( x-c-\varrho \frac{d-c}{2}
\right) ^{3}\right) \right] .
\end{align}
(23)
Proof.
Let us define the mapping
\begin{equation}
L\left( x,t\right) =\left\{
\begin{array}{ccc}
\frac{\varphi }{2\left( \varphi +\psi \right) }\left[ t-\left( c+\varrho
\frac{d-c}{2}\right) \right] ^{2}, & & \ \ t\in \left[ c,x\right] \\
\frac{\psi }{2\left( \varphi +\psi \right) }\left[ t-\left( d-\varrho \frac{
d-c}{2}\right) \right] ^{2}, & & \ \ t\in (x,d]
\end{array}
\right. \label{M-3}
\end{equation}
(24)
By using (24), we get
\begin{align}
\label{3.46}
\int\limits_{c}^{d}L\left( x,t\right) z^{\prime \prime }\left( t\right) dt
=&\frac{1}{\varphi +\psi }\left[ \frac{\varphi }{2}\left[ x-\left(
c+\varrho \frac{d-c}{2}\right) \right] ^{2}z^{\prime }\left( x\right) -\frac{
\psi }{2}\right. \left[ x-\left( d-\varrho \frac{d-c}{2}\right) \right] ^{2}z^{\text{
}\prime }\left( x\right) -\frac{\varphi }{8}\varrho ^{2}\left( d-c\right)
^{2}z^{\prime }\left( c\right) \notag \\
& +\frac{\psi }{8}\varrho ^{2}\left( d-c\right) ^{2}z^{\prime }\left(
d\right) -\varphi \left[ x-\left( c+\varrho \frac{d-c}{2}\right) \right] z\left(
x\right) +\psi \left[ x-\left( d-\varrho \frac{d-c}{2}\right) \right] z\left( x\right)\notag
\\
& -\frac{\varphi }{2}\varrho \left( d-c\right)
z\left( c\right) -\frac{\psi }{2}\varrho \left( d-c\right) z\left( d\right)
\left. +\varphi \int\limits_{c}^{x}z\left( t\right) dt+\psi
\int\limits_{x}^{d}z\left( t\right) dt\right]
\end{align}
(25)
and
\begin{align}
\label{3.47}
\int\limits_{c}^{d}L\left( x,t\right) dt =\frac{1}{6\left( \varphi +\psi \right) }\left( \varphi \left( x-c-\varrho
\frac{d-c}{2}\right) ^{3}\ -\psi \left( x-d+\varrho \frac{d-c}{2}\right)
^{3}\right)
& +\frac{\varrho ^{3}}{48}\left( d-c\right) ^{3} .
\end{align}
(26)
Using (25) and (26), we get
\begin{align}
& \int\limits_{c}^{d}L\left( x,t\right) \left( z^{\prime \prime }\left(
t\right) -C\right) dt =\frac{1}{\varphi +\psi }\left[ \frac{\varphi }{2}\left[ x-\left(
c+\varrho \frac{d-c}{2}\right) \right] ^{2}z^{\prime }\left( x\right) \right.
-\frac{\psi }{2}\left[ x-\left( d-\varrho \frac{d-c}{2}\right) \right]
^{2}z^{\prime }\left( x\right)\notag\\
& -\frac{\varphi }{8}\varrho ^{2}\left( d-c\right) ^{2}z^{\text{ }\prime
}\left( c\right) +\frac{\psi }{8}\varrho ^{2}\left( d-c\right) ^{2}z^{\prime
}\left( d\right) -\varphi \left[ x-\left( c+\varrho \frac{d-c}{2}\right) \right] z\left(
x\right) +\psi \left[ x-\left( d-\varrho \frac{d-c}{2}\right) \right]
z\left( x\right) \notag\\
\label{3.48}
& -\frac{\varphi }{2}\varrho \left( d-c\right) z\left( c\right) -\frac{\psi
}{2}\varrho \left( d-c\right) z\left( d\right) \left. +\varphi \int\limits_{c}^{x}z\left( t\right) dt+\psi
\int\limits_{x}^{d}z\left( t\right) dt\right] -C\left[ \frac{\varrho ^{3}}{
48}\left( d-c\right) ^{3}+\frac{1}{6\left( \varphi +\psi \right) }\right.
\notag \\
& \left. \times \left( \varphi \left( x-c-\varrho \frac{d-c}{2}\right) ^{3}\
-\psi \left( x-d+\varrho \frac{d-c}{2}\right) ^{3}\right) \right] .
\end{align}
(27)
But on the other side,
\begin{equation}
\left\vert \int\limits_{c}^{d}L\left( x,t\right) \left( z^{\prime \prime
}\left( t\right) -C\right) dt\right\vert \leq \overset{}{\underset{t\in
\left[ c,d\right] }{\max }}\left\vert z^{\prime \prime }\left( t\right)
-C\right\vert \int\limits_{c}^{d}\left\vert L\left( x,t\right) \right\vert
dt. \label{3.49}
\end{equation}
(28)
Now, again by using (24), we get
\begin{align}
\label{3.50}
\int\limits_{c}^{d}\left\vert L\left( x,t\right) \right\vert dt
=-\frac{\varrho ^{3}\left( \varphi -\psi \right) }{48\left( \varphi +\psi
\right) }\left( d-c\right) ^{3}+\frac{1}{6\left( \varphi +\psi \right) }
\left( \varphi \left[ x-\left( c+\varrho \frac{d-c}{2}\right)
\right] ^{3}+\psi \left[ x-\left( d-\varrho \frac{d-c}{2}\right) \right]
^{3}\right)
\end{align}
(29)
and
\begin{equation}
C=\frac{\Gamma +\gamma }{2}. \label{3.51}
\end{equation}
(30)
Also
\begin{equation}
\overset{}{\underset{t\in \left[ c,d\right] }{\max }}\left\vert z^{\prime
\prime }\left( t\right) -C\right\vert \leq \frac{\Gamma -\gamma }{2}.
\label{3.52}
\end{equation}
(31)
Using (28) and (29), we get
\begin{align}
\label{3.53}
&\left\vert \int\limits_{c}^{d}L\left( x,t\right) \left( z^{\prime \prime
}\left( t\right) -C\right) dt\right\vert \notag\\
& =-\frac{\varrho ^{3}\left( \varphi -\psi \right) }{48\left( \varphi +\psi
\right) }\left( d-c\right) ^{3}+\frac{1}{6\left( \varphi +\psi \right) }
\left( \varphi \left[ x-\left( c+\varrho \frac{d-c}{2}\right)
\right] ^{3}+\psi \left[ x-\left( d-\varrho \frac{d-c}{2}\right) \right]
^{3}\right) .
\end{align}
(32)
Using (27) and (32), we get our required result (23).
6. Conclusion
In this paper, we proved the results by using quadratic mapping, generalized
linear mapping and generalized quadratic mapping. We developed application
for numerical integration also.
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflicts of Interest
“The authors declare no conflict of interest”.
