Engineering and Applied Science Letter
Vol. 4 (2021), Issue 1, pp. 14 – 20
ISSN: 2617-9709 (Online) 2617-9695 (Print)
DOI: 10.30538/psrp-easl2021.0057
ISSN: 2617-9709 (Online) 2617-9695 (Print)
DOI: 10.30538/psrp-easl2021.0057
Čebyšev inequalities for co-ordinated \(QC\)-convex and \((s,QC)\)-convex
B. Meftah\(^1\), A. Souahi
Laboratoire des Télécommunications, Faculté des Sciences et de la Technologie, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria; (B.M)
Laboratory of Advanced Materials, University of Badji Mokhtar-Annaba, P.O. Box 12, 23000 Annaba, Algeria.; (A.S)
\(^{1}\)Corresponding Author: badrimeftah@yahoo.fr
Copyright © 2021 B. Meftah, A. Souahi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: July 19, 2020 – Accepted: January 1, 2021 – Published: January 23, 2021
Abstract
In this paper, we establish some new Čebyšev type inequalities for functions whose modulus of the mixed derivatives are co-ordinated quasi-convex and \(\alpha\)-quasi-convex and \(s\)-quasi-convex functions.
Keywords:
Čebyšev inequalities, quasi-convexity, \((s,QC)\)-convexity, \((\alpha ,QC)\)-convexity.
1. Introduction
In 1882, Cebyšev [1] gave the following inequality
\begin{equation}
\label{1.1}
\left\vert T\left( f,g\right) \right\vert \leq \tfrac{1}{12}\left(
b-a\right) ^{2}\left\Vert f^{\prime }\right\Vert _{\infty }\left\Vert
g^{\prime }\right\Vert _{\infty },
\end{equation}
(1)
\begin{equation}
\label{1.2}
T\left( f,g\right) =\tfrac{1}{b-a}\overset{b}{\underset{a}{\int }}f\left(
x\right) g\left( x\right) dx-\left( \tfrac{1}{b-a}\overset{b}{\underset{a}{
\int }}f\left( x\right) dx\right) \left( \tfrac{1}{b-a}\overset{b}{\underset{
a}{\int }}g\left( x\right) dx\right) ,
\end{equation}
(2)
During the past few years, many researchers have given considerable attention to the inequality (1). Various generalizations, extensions and variants have been appeared in the literature [2,3,4,5,6].
Recently, Guezane-Lakoud and Aissaoui [2] gave the analogue of the functional (2) for functions of two variables and established the following Cebyšev type inequalities for functions whose mixed derivatives are bounded as follows;
\begin{equation}
\label{1.3}
\left\vert T(f,g)\right\vert \leq \tfrac{49}{3600}k^{2}\left\Vert
f_{_{\lambda \alpha }}\right\Vert _{\infty }\left\Vert g_{_{\lambda \alpha
}}\right\Vert _{\infty },
\end{equation}
(3)
\begin{align}
\left\vert T(f,g)\right\vert \leq \frac{1}{8k^{2}}\overset{b}{\underset{a}
{\int }}\overset{d}{\underset{c}{\int }}\left[ \left( \left\vert
g(x,y)\right\vert \left\Vert f_{_{\lambda \alpha }}\right\Vert _{\infty
}+\left\vert f(x,y)\right\vert \left\Vert g_{_{\lambda \alpha }}\right\Vert
_{\infty }\right) \right. \left. \left[ \left( \left( x-a\right) ^{2}+\left( b-x\right)
^{2}\right) \left( \left( y-c\right) ^{2}+\left( d-y\right) ^{2}\right)
\right] \right] dydx, \label{1.4}
\end{align}
(4)
\begin{align}
T(f,g) =&\tfrac{1}{k}\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d
}{\int }}f\left( x,y\right) g\left( x,y\right) dydx-\tfrac{d-c}{k^{2}\ }
\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}{\int }}g\left(
x,y\right) \left( \underset{a}{\overset{b}{\int }}f\left( t,y\right)
dt\right) dydx \notag\\
&-\tfrac{b-a}{k^{2}}\underset{a}{\overset{b}{\int }}\underset{c}{\overset{d}
{\int }}g\left( x,y\right) \left( \underset{c}{\overset{d}{\int }}f\left(
x,v\right) dv\right) dydx +\tfrac{1}{k^{2}\ }\left( \underset{a}{\overset{b}{\int }}\underset{c}{
\overset{d}{\int }}f\left( x,y\right) dydx\right) \left( \underset{a}{
\overset{b}{\int }}\underset{c}{\overset{d}{\int }}g\left( t,v\right)
dvdt\right) . \label{1.5}
\end{align}
(5)
2. Preliminaries
Throughout this paper, we denote by \(\Delta \), the bidimensional interval in \(% [0,\infty )^{2}\), \(\Delta =:[a,b]\times \lbrack c,d]\) with \(a< b\) and \(c< d\), \(% k=:\left( b-a\right) (d-c)\) and \(\frac{\partial ^{2}f}{\partial \lambda \partial w}\) by \(f_{\lambda w}.\)Definition 1. [7] A function \(f:\Delta \rightarrow \mathbb{R}\) is said to be convex on the co-ordinates on \(\Delta \) if \begin{align*} f\left( \lambda x+\left( 1-\lambda \right) t,wy+\left( 1-w\right) v\right) \leq &\lambda wf(x,y)+\lambda \left( 1-w\right) f(x,v)+\left( 1-\lambda \right) wf(t,y)+\left( 1-\lambda \right) \left( 1-w\right) f(t,v) \end{align*} holds for all \(\lambda ,w\in \lbrack 0,1]\) and \((x,y),(x,v),(t,y),(t,v)\in \Delta \).
Definition 2. [8] A function \(f:\Delta \rightarrow \mathbb{R}\) is said to be quasi-convex on the co-ordinates on \(\Delta \) if \begin{equation*} f\left( \lambda x+\left( 1-\lambda \right) t,wy+\left( 1-w\right) v\right) \leq \max \left\{ f(x,y)+f(x,v)+f(t,y)+f(t,v)\right\} \end{equation*} holds for all \(\lambda ,w\in \lbrack 0,1]\) and \((x,y),(x,v),(t,y),(t,v)\in \Delta \).
Definition 3. [9] For some \(\alpha \in \left( 0,1\right] \), a function \(f:\Delta \rightarrow \mathbb{R}\) is said to be \(\left( \alpha ,QC\right) \)-convex on the co-ordinates on \(\Delta \), if \begin{align*} f\left( \lambda x+\left( 1-\lambda \right) t,wy+\left( 1-w\right) v\right) \leq &\lambda ^{\alpha }\max \left\{ f(x,y)+f(x,v)\right\} +\left( 1-\lambda ^{\alpha }\right) \max \left\{ f(t,y)+f(t,v)\right\} \end{align*} holds for all \(\lambda ,w\in \lbrack 0,1]\) and \((x,y),(x,v),(t,y),(t,v)\in \Delta \).
Definition 4. [10] For some \(s\in \left[ -1,1\right] \), a function \(f:\Delta \rightarrow \left[ 0,\infty \right) \) is said to be \(\left( s,QC\right) \)-convex on co-ordinates on \(\Delta \), if \begin{align*} f\left( \lambda x+\left( 1-\lambda \right) t,wy+\left( 1-w\right) v\right) \leq &\lambda ^{s}\max \left\{ f(x,y)+f(x,v)\right\} +\left( 1-\lambda \right) ^{s}\max \left\{ f(t,y)+f(t,v)\right\} \end{align*} holds for all \(\lambda \in \left( 0,1\right),\ w\in \lbrack 0,1]\) and \(% (x,y),(x,v),(t,y),(t,v)\in \Delta \).
Lemma 1. [11] Let f : \(\Delta \ \rightarrow \mathbb{R}\) be a partial differentiable mapping on \(\Delta \) in \(\mathbb{R}^{2}\). If \(f_{\lambda w}\in L_{1}\left( \Delta \ \right) \) then for any \(\left( x,y\right) \in \Delta \), we have the equality;
\begin{align}
f(x,y) =&\tfrac{1}{b-a}\overset{b}{\underset{a}{\int }}f(t,y)dt+\tfrac{1}{
d-c}\overset{d}{\underset{c}{\int }}f(x,v)dv-\tfrac{1}{k}\overset{b}{
\underset{a}{\int }}\overset{d}{\underset{c}{\int }}f(t,v)dvdt +\tfrac{1}{k}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int
}}\left( x-t\right) \left( y-v\right) \notag\\
&\times \left( \overset{1}{\underset{0}{\int }}\overset{1}{\underset{0}{
\int }}f_{_{\lambda w}}\left( \lambda x+(1-\lambda )t,wy-(1-w)v\right)
dwd\lambda \right) dvdt. \label{2.1}
\end{align}
(6)
3. Main result
Theorem 1. Let \(f,g\) \(:\Delta \ \rightarrow \mathbb{R}\) be partially differentiable functions such that their second derivatives \(f_{\lambda w}\) and \(% g_{\lambda w}\) are integrable on \(\Delta \). If \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \) are co-ordinated quasi-convex on \(\Delta \), then
\begin{equation}
\label{3.1}
\left\vert T(f,g)\right\vert \leq \tfrac{49}{3600}MNk^{2},
\end{equation}
(7)
Proof. From Lemma 1, we have
\begin{align}
f(x,y)&-\tfrac{1}{b-a}\overset{b}{\underset{a}{\int }}f(t,y)dt-\tfrac{1}{d-c
}\overset{d}{\underset{c}{\int }}f(x,v)dv+\tfrac{1}{k}\overset{b}{\underset{a
}{\int }}\overset{d}{\underset{c}{\int }}f(t,v)dvdt \notag\\
=&\tfrac{1}{k}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int
}}\left( x-t\right) \left( y-v\right) \left( \overset{1}{\underset{0}{\int }}
\overset{1}{\underset{0}{\int }}f_{_{\lambda w}}\left( \lambda x+(1-\lambda
)t,wy-(1-w)v\right) d\alpha d\lambda \right) dvdt, \label{3.2}
\end{align}
(8)
\begin{align}
g(x,y)&-\tfrac{1}{b-a}\overset{b}{\underset{a}{\int }}g(t,y)dt-\tfrac{1}{d-c
}\overset{d}{\underset{c}{\int }}g(x,v)dv+\tfrac{1}{k}\overset{b}{\underset{a
}{\int }}\overset{d}{\underset{c}{\int }}g(t,v)dvdt \notag\\
=&\tfrac{1}{k}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int
}}\left( x-t\right) \left( y-v\right) \left( \overset{1}{\underset{0}{\int }}
\overset{1}{\underset{0}{\int }}g_{_{\lambda w}}\left( \lambda x+(1-\lambda
)t,wy-(1-w)v\right) dwd\lambda \right) dvdt. \label{3.3}
\end{align}
(9)
\begin{align}
\left\vert T(f,g)\right\vert \leq &\tfrac{1}{k^{3}}\overset{b}{\underset{a}{
\int }}\overset{d}{\underset{c}{\int }}\left[ \overset{b}{\underset{a}{\int }
}\overset{d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert
y-v\right\vert \right. \times \left. \left( \overset{1}{\underset{0}{\int }}\overset{1}{\underset{
0}{\int }}\left\vert f_{_{\lambda w}}\left( \lambda x+(1-\lambda
)t,wy-(1-w)v\right) \right\vert dwd\lambda \right) dvdt\right] \notag\\
&\times \left[ \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert \right. \times \left. \left. \left( \overset{1}{\underset{0}{\int }}\overset{1}{
\underset{0}{\int }}g_{_{\lambda w}}\left\vert \left( \lambda x+(1-\lambda
)t,wy-(1-w)v\right) \right\vert dwd\lambda \right) \right] dvdt\right] dydx.
\label{3.4}
\end{align}
(10)
\begin{align}
\left\vert T(f,g)\right\vert \leq &\tfrac{1}{k^{3}}MN\overset{b}{\underset{a
}{\int }}\overset{d}{\underset{c}{\int }}\left( \overset{b}{\underset{a}{
\int }}\overset{d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert
y-v\right\vert dvdt\right) ^{2}dydx
=\tfrac{49}{3600}k^{2}MN, \label{3.5}
\end{align}
(11)
\begin{equation}
\label{3.6}
\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }}\left(
\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }}\left\vert
x-t\right\vert \left\vert y-v\right\vert dvdt\right) ^{2}dydx=\tfrac{49}{3600
}k^{5}.
\end{equation}
(12)
Theorem 2. Under the assumptions of Theorem 1, we have
\begin{align}
\left\vert T(f,g)\right\vert \leq &\tfrac{1}{8k^{2}}\left[ \overset{b}{
\underset{a}{\int }}\overset{d}{\underset{c}{\int }}\left[ M\left\vert
g(x,y)\right\vert +N\left\vert f(x,y)\right\vert \right] \left[ \left(
x-a\right) ^{2}+\left( b-x\right) ^{2}\right] \right. \times \left. \ \left[ \left( y-c\right) ^{2}+\left( d-y\right) ^{2}\right]
\right] dydx, \label{3.7}
\end{align}
(13)
Proof. From Lemma 1, (8) and (9) are valid. Let \(G(x,y)=\frac{1}{2k}g(x,y)\) and \(F(x,y)=\frac{1}{2k}f(x,y)\). Multiplying \(G(x,y)\) by \(F(x,y)\), then integrating the resultant equalities with respect to \(x\) and \(y\) over \(% \Delta \), and by using the modulus, we get
\begin{align}
\left\vert T(f,g)\right\vert \leq& \tfrac{1}{2k^{2}}\left[ \overset{b}{
\underset{a}{\int }}\overset{d}{\underset{c}{\int }}\left\vert
g(x,y)\right\vert \left[ \overset{b}{\underset{a}{\int }}\overset{d}{
\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\right. \right. \notag\\&\times \left. \left( \overset{1}{\underset{0}{\int }}\overset{1}{\underset{
0}{\int }}\left\vert f_{_{\lambda w}}\left( \lambda x+(1-\lambda
)t,wy-(1-w)v\right) \right\vert dwd\lambda \right) dvdt\right] dydx+\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }}
\left\vert f(x,y)\right\vert \left[ \overset{b}{\underset{a}{\int }}\overset{
d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\right. \notag\\
& \times \left. \left( \overset{1}{\underset{0}{\int }}\overset{1}{\underset{
0}{\int }}\left\vert g_{_{\lambda w}}\left( \lambda x+(1-\lambda
)t,wy-(1-w)v\right) \right\vert dwd\lambda \right) dvdt\right] dydx. \label{3.8}
\end{align}
(14)
\begin{align}
\left\vert T(f,g)\right\vert \leq& \tfrac{1}{2k^{2}}\left[ \overset{b}{
\underset{a}{\int }}\overset{d}{\underset{c}{\int }}M\left\vert
g(x,y)\right\vert \left( \overset{b}{\underset{a}{\int }}\overset{d}{
\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
dvdt\right) dydx\right. \notag\\&+\left. \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }}
N\left\vert f(x,y)\right\vert \left( \overset{b}{\underset{a}{\int }}\overset
{d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
dvdt\right) \right] dydx \notag\\
=&\tfrac{1}{2k^{2}}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left( M\left\vert g(x,y)\right\vert +N\left\vert f(x,y)\right\vert
\right) \left( \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int
}}\left\vert x-t\right\vert \left\vert y-v\right\vert dvdt\right) dydx.
\label{3.9}
\end{align}
(15)
\begin{align}
\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }}\left\vert
x-t\right\vert \left\vert y-v\right\vert dvdt =&\tfrac{1}{4}\left[ \left(
x-a\right) ^{2}+\left( b-x\right) ^{2}\right] \left[ \left( y-c\right)
^{2}+\left( d-y\right) ^{2}\right] . \label{3.10}
\end{align}
(16)
Theorem 3. Let \(f,g\) \(:\Delta \ \rightarrow \mathbb{R}\) be partially differentiable functions, such that their second derivatives \(f_{\lambda w}\) and \(% g_{\lambda w}\) are integrable on \(\Delta \). If \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \) are co-ordinated \(% \alpha \)-quasi-convex on \(\Delta \), for some \(\alpha \in \left( 0,1\right] \) , then
\begin{equation}
\label{3.11}
\left\vert T(f,g)\right\vert \leq \tfrac{49}{3600}MNk^{2},
\end{equation}
(17)
Proof. Clearly the inequalities (8)-(10) are valid, using the co-ordinated \(% \alpha \)-quasi-convexity of \(\left\vert f_{\lambda w}\right\vert \) and \(% \left\vert g_{\lambda w}\right\vert \), (10) gives
\begin{align*}
\left\vert T(f,g)\right\vert \leq &\tfrac{1}{k^{3}}\overset{b}{\underset{a}{
\int }}\overset{d}{\underset{c}{\int }}\left[ \overset{b}{\underset{a}{\int }
}\overset{d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert
y-v\right\vert \overset{1}{\underset{0}{\int }}\overset{1}{\underset{0}{\int
}}\left[ \lambda ^{\alpha }\max \left\{ \left\vert f_{_{\lambda
w}}(x,y)\right\vert +\left\vert f_{_{\lambda w}}(x,v)\right\vert \right\}
\right. \right. \notag\\
&+\left. \left. \left. \left( 1-\lambda ^{\alpha }\right) \max \left\{
\left\vert f_{_{\lambda w}}(t,y)\right\vert +\left\vert f_{_{\lambda
w}}(t,v)\right\vert \right\} \right] dwd\lambda \right) dvdt\right] \notag\\
&\times \left[ \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert \overset{1}{
\underset{0}{\int }}\overset{1}{\underset{0}{\int }}\left[ \lambda ^{\alpha
}\max \left\{ \left\vert g_{_{\lambda w}}(x,y)\right\vert +\left\vert
g_{_{\lambda w}}(x,v)\right\vert \right\} \right. \right. \notag\\
&+\left. \left. \left. \left( 1-\lambda ^{\alpha }\right) \max \left\{
\left\vert g_{_{\lambda w}}(t,y)\right\vert +\left\vert g_{_{\lambda
w}}(t,v)\right\vert \right\} \right] dwd\lambda \right] dvdt\right] dydx\\
\notag
=&\tfrac{1}{k^{3}}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left[ \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }
}\left\vert x-t\right\vert \left\vert y-v\right\vert \left[ \max \left\{
\left\vert f_{_{\lambda w}}(x,y)\right\vert +\left\vert f_{_{\lambda
w}}(x,v)\right\vert \right\} \overset{1}{\underset{0}{\int }}\overset{1}{
\underset{0}{\int }}\lambda ^{\alpha }dwd\lambda \right. \right.
\end{align*}
\begin{align}
\notag
&+\left. \left. \max \left\{ \left\vert f_{_{\lambda w}}(t,y)\right\vert
+\left\vert f_{_{\lambda w}}(t,v)\right\vert \right\} \overset{1}{\underset{0
}{\int }}\overset{1}{\underset{0}{\int }}\left( 1-\lambda ^{\alpha }\right)
dwd\lambda \right] dvdt\right] \notag\\
&\times \left[ \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert \left[ \max
\left\{ \left\vert g_{_{\lambda w}}(x,y)\right\vert +\left\vert g_{_{\lambda
w}}(x,v)\right\vert \right\} \overset{1}{\underset{0}{\int }}\overset{1}{
\underset{0}{\int }}\lambda ^{\alpha }dwd\lambda \right. \right. \notag\\
&+\left. \left. \max \left\{ \left\vert g_{_{\lambda w}}(t,y)\right\vert
+\left\vert g_{_{\lambda w}}(t,v)\right\vert \right\} \overset{1}{\underset{0
}{\int }}\overset{1}{\underset{0}{\int }}\left( 1-\lambda ^{\alpha }\right)
dwd\lambda \right] dvdt\right] dydx \notag\\
\leq &\tfrac{1}{k^{3}}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{
c}{\int }}\left[ \left[ \left( \overset{b}{\underset{a}{\int }}\overset{d}{
\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\left( \tfrac{1}{\alpha +1}+1-\tfrac{1}{\alpha +1}\right) Mdvdt\right)
\right] \right. \notag\\
&\times \left. \left[ \left( \overset{b}{\underset{a}{\int }}\overset{d}{
\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\left( \tfrac{1}{\alpha +1}+1-\tfrac{1}{\alpha +1}\right) Ndvdt\right)
\right] \right] dydx \notag\\
=&\tfrac{MN}{k^{3}}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left( \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }
}\left\vert x-t\right\vert \left\vert y-v\right\vert \right) ^{2}dydx. \label{3.12}
\end{align}
(18)
Theorem 4. Under the assumptions of Theorem 3, we have
\begin{align}
\left\vert T(f,g)\right\vert \leq &\tfrac{1}{8k^{2}}\left[ \overset{b}{
\underset{a}{\int }}\overset{d}{\underset{c}{\int }}\left( M\left\vert
g(x,y)\right\vert +N\left\vert f(x,y)\right\vert \right) \right. \times \left[ \left( x-a\right) ^{2}+\left( b-x\right) ^{2}\right] \left[
\left( y-c\right) ^{2}+\left( d-y\right) ^{2}\right] dydx, \label{3.13}
\end{align}
(19)
Proof. By the same argument given in Theorem 2, we easly obtain the inequality (14), using the \(\alpha \)-quasi-convexity on the co-ordinates of \(% \left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \), we get
\begin{align}
\left\vert T(f,g)\right\vert \leq &\tfrac{1}{2k^{2}}\left[ \overset{b}{
\underset{a}{\int }}\overset{d}{\underset{c}{\int }}\left\vert
g(x,y)\right\vert \left[ \overset{b}{\underset{a}{\int }}\overset{d}{
\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\right. \right. \times \left. \left( M\overset{1}{\underset{0}{\int }}\overset{1}{\underset
{0}{\int }}\lambda ^{\alpha }dwd\lambda +M\overset{1}{\underset{0}{\int }}
\overset{1}{\underset{0}{\int }}\left( 1-\lambda ^{\alpha }\right)
dwd\lambda \right) dvdt\right] dydx \notag\\
&\ +\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }}
\left\vert f(x,y)\right\vert \left[ \overset{b}{\underset{a}{\int }}\overset{
d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\right. \times \left. \left( N\overset{1}{\underset{0}{\int }}\overset{1}{\underset
{0}{\int }}\lambda ^{\alpha }dwd\lambda +N\overset{1}{\underset{0}{\int }}
\overset{1}{\underset{0}{\int }}\left( 1-\lambda ^{\alpha }\right)
dwd\lambda \right) dvdt\right] dydx. \notag\\
=&\tfrac{1}{2k^{2}}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left[ \left( M\left\vert g(x,y)\right\vert +N\left\vert
f(x,y)\right\vert \right) \overset{b}{\underset{a}{\int }}\overset{d}{
\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert dvdt
\right] dydx. \label{3.14}
\end{align}
(20)
Theorem 5. Let \(f,g\) \(:\Delta \ \rightarrow \mathbb{R}\) be partially differentiable functions such that their second derivatives \(f_{\lambda w}\) and \(% g_{\lambda w}\) are integrable on \(\Delta \), and let \(s\in \left( -1,1\right] \) fixed. If \(\left\vert f_{\lambda \alpha }\right\vert \) and \(\left\vert g_{\lambda \alpha }\right\vert \) are co-ordinated \(s\)-quasi-convex on \(% \Delta \), then
\begin{equation}
\label{3.15}
\left\vert T(f,g)\right\vert \leq \tfrac{49}{900\left( s+1\right) ^{2}}
MNk^{2},
\end{equation}
(21)
Proof. Clearly inequalities (8)-(10) are satisfied. Using second definition of the co-ordinated \(s\)-quasi-convex of \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \), (10) gives;
\begin{align}
\left\vert T(f,g)\right\vert \leq &\tfrac{1}{k^{3}}\overset{b}{\underset{a}{
\int }}\overset{d}{\underset{c}{\int }}\left[ \overset{b}{\underset{a}{\int }
}\overset{d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert
y-v\right\vert \overset{1}{\underset{0}{\int }}\overset{1}{\underset{0}{\int
}}\left[ \lambda ^{s}\max \left\{ \left\vert f_{_{\lambda
w}}(x,y)\right\vert +\left\vert f_{_{\lambda w}}(x,v)\right\vert \right\}
\right. \right. \notag\\
&+\left. \left. \left. \left( 1-\lambda \right) ^{s}\max \left\{ \left\vert
f_{_{\lambda w}}(t,y)\right\vert +\left\vert f_{_{\lambda
w}}(t,v)\right\vert \right\} \right] dwd\lambda \right) dvdt\right] \notag\\
&\times \left[ \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert \overset{1}{
\underset{0}{\int }}\overset{1}{\underset{0}{\int }}\left[ \lambda ^{s}\max
\left\{ \left\vert g_{_{\lambda w}}(x,y)\right\vert +\left\vert g_{_{\lambda
w}}(x,v)\right\vert \right\} \right. \right. \notag\\
&+\left. \left. \left. \left( 1-\lambda \right) ^{s}\max \left\{ \left\vert
g_{_{\lambda w}}(t,y)\right\vert +\left\vert g_{_{\lambda
w}}(t,v)\right\vert \right\} \right] dwd\lambda \right] dvdt\right] dydx
\notag\\
=&\tfrac{1}{k^{3}}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left[ \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }
}\left\vert x-t\right\vert \left\vert y-v\right\vert \left[ \max \left\{
\left\vert f_{_{\lambda w}}(x,y)\right\vert +\left\vert f_{_{\lambda
w}}(x,v)\right\vert \right\} \overset{1}{\underset{0}{\int }}\overset{1}{
\underset{0}{\int }}\lambda ^{s}dwd\lambda \right. \right. \notag\\
&+\left. \left. \max \left\{ \left\vert f_{_{\lambda w}}(t,y)\right\vert
+\left\vert f_{_{\lambda w}}(t,v)\right\vert \right\} \overset{1}{\underset{0
}{\int }}\overset{1}{\underset{0}{\int }}\left( 1-\lambda \right)
^{s}dwd\lambda \right] dvdt\right] \notag\\
&\times \left[ \overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{
\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert \left[ \max
\left\{ \left\vert g_{_{\lambda w}}(x,y)\right\vert +\left\vert g_{_{\lambda
w}}(x,v)\right\vert \right\} \overset{1}{\underset{0}{\int }}\overset{1}{
\underset{0}{\int }}\lambda ^{s}dwd\lambda \right. \right. \notag\\
&+\left. \left. \max \left\{ \left\vert g_{_{\lambda w}}(t,y)\right\vert
+\left\vert g_{_{\lambda w}}(t,v)\right\vert \right\} \overset{1}{\underset{0
}{\int }}\overset{1}{\underset{0}{\int }}\left( 1-\lambda \right)
^{s}dwd\lambda \right] dvdt\right] dydx \notag\\
\leq &\frac{1}{k^{3}}\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c
}{\int }}\left[ \left[ \left( \overset{b}{\underset{a}{\int }}\overset{d}{
\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\left( \tfrac{M}{s+1}+\tfrac{M}{s+1}\right) dvdt\right) \right] \right.
\notag\\
&\times \left. \left[ \left( \overset{b}{\underset{a}{\int }}\overset{d}{
\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\left( \tfrac{N}{s+1}+\tfrac{N}{s+1}\right) dvdt\right) \right] \right] dydx
\notag\\
=&\tfrac{4MN}{\left( s+1\right) ^{2}k^{3}}\overset{b}{\underset{a}{\int }}
\overset{d}{\underset{c}{\int }}\left( \overset{b}{\underset{a}{\int }}
\overset{d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert
y-v\right\vert \right) ^{2}dydx. \label{3.16}
\end{align}
(22)
Theorem 6. Under the assumptions of Theorem 5, we have
\begin{align}
\left\vert T(f,g)\right\vert \leq &\tfrac{1}{4\left( s+1\right) k^{2}}\left[
\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }}\left(
M\left\vert g(x,y)\right\vert +N\left\vert f(x,y)\right\vert \right) \right.
\times \left[ \left( x-a\right) ^{2}+\left( b-x\right) ^{2}\right] \left[
\left( y-c\right) ^{2}+\left( d-y\right) ^{2}\right] dydx, \label{3.17}
\end{align}
(23)
Proof. By the same argument given in Theorem 2, we easily obtain the inequality (14), using the second definition of \(s\)-quasi-convexity on the co-ordinates of \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda w}\right\vert \), we get
\begin{align}
\left\vert T(f,g)\right\vert \leq &\tfrac{1}{2k^{2}}\left[ \overset{b}{
\underset{a}{\int }}\overset{d}{\underset{c}{\int }}\left\vert
g(x,y)\right\vert \left[ \overset{b}{\underset{a}{\int }}\overset{d}{
\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\right. \right. \times \left. \left( M\overset{1}{\underset{0}{\int }}\overset{1}{\underset
{0}{\int }}\lambda ^{s}dwd\lambda +M\overset{1}{\underset{0}{\int }}\overset{
1}{\underset{0}{\int }}\left( 1-\lambda \right) ^{s}dwd\lambda \right) dvdt
\right] dydx \notag\\
&\ +\overset{b}{\underset{a}{\int }}\overset{d}{\underset{c}{\int }}
\left\vert f(x,y)\right\vert \left[ \overset{b}{\underset{a}{\int }}\overset{
d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert y-v\right\vert
\right. \times \left. \left( N\overset{1}{\underset{0}{\int }}\overset{1}{\underset
{0}{\int }}\lambda ^{s}dwd\lambda +N\overset{1}{\underset{0}{\int }}\overset{
1}{\underset{0}{\int }}\left( 1-\lambda \right) ^{s}dwd\lambda \right) dvdt
\right] dydx. \notag\\
=&\tfrac{1}{\left( s+1\right) k^{2}}\overset{b}{\underset{a}{\int }}\overset
{d}{\underset{c}{\int }}\left[ \left( M\left\vert g(x,y)\right\vert
+N\left\vert f(x,y)\right\vert \right) \overset{b}{\underset{a}{\int }}
\overset{d}{\underset{c}{\int }}\left\vert x-t\right\vert \left\vert
y-v\right\vert dvdt\right] dydx. \label{3.18}
\end{align}
(24)
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.Conflicts of Interests
The authors declare no conflict of interest.References
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