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)
where \(f,g:\left[ a,b\right] \rightarrow \mathbb{R}\) are absolutely
continuous function, whose first derivatives \(f^{\prime }\) and \(g^{\prime }\)
are bounded and
\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)
and \(\left\Vert .\right\Vert _{\infty }\) denotes the norm in \(L_{\infty }%
\left[ a,b\right] \) defined as \(\left\Vert f\right\Vert _{\infty }=\underset{%
t\in \left[ a,b\right] }{ess\sup }\left\vert f\left( t\right) \right\vert \).
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)
and
\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)
where
\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)
Motivated by the existing results, in this paper we establish some new Cebyšev type inequalities for functions whose mixed derivatives are
co-ordinates quasi-convex and co-ordinates \((\alpha ,QC)\) and \((s,QC)\)
-convex.
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)
where \(T(f,g)\) is defined as in (5),
\(M=\underset{x,t\in \left[ a,b\right] ,y,v\in \left[ c,d\right] }{\max }\) \(%
\left[ \left\vert f_{\lambda w}\left( x,y\right) \right\vert +\left\vert
f_{\lambda w}\left( x,v\right) \right\vert +\left\vert f_{\lambda w}\left(
t,y\right) \right\vert +\left\vert f_{\lambda w}\left( t,v\right)
\right\vert \right] \),
and
\(N=\underset{x,t\in \left[ a,b\right] ,y,v\in \left[ c,d\right] }{\max }%
\left[ \left\vert g_{\lambda w}\left( x,y\right) \right\vert +\left\vert
g_{\lambda w}\left( x,v\right) \right\vert +\left\vert g_{\lambda w}\left(
t,y\right) \right\vert +\left\vert g_{\lambda w}\left( t,v\right)
\right\vert \right] \), and \(k=\left( b-a\right) \left( d-c\right) \).
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)
and
\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)
Multiplying (8) by (9), and then integrating the resulting equality
with respect to \(x\) and \(y\) over \(\Delta \), using modulus and Fubini’s
Theorem, and multiplying the result by \(\frac{1}{k}\), we get
\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)
Since \(\left\vert f_{\lambda \alpha }\right\vert \) and \(\left\vert
g_{\lambda \alpha }\right\vert \) are co-ordinated quasi-convex, we deduce
\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)
where we have used the fact that
\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)
The proof is completed.
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)
where \(T(f,g)\) is defined as in (5), \(M,\) \(N,\) and \(k\) are as in Theorem 1.
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)
Since \(\left\vert f_{\lambda w}\right\vert \) and \(\left\vert g_{\lambda
w}\right\vert \) are co-ordinated quasi-convex, (14) implies
\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)
By a simple computation, we easily obtain
\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)
Substituting (16) in (15), we get the desired result.
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)
where \(T(f,g)\) is defined as in (5), \(M,N\), and \(k\) are as in Theorem 1.
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)
Using (12) in (18), we obtain the desired result.
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)
where \(T(f,g)\) is defined as in (5) and \(M,N,\) and \(k\) are as in Theorem 3.
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)
Substituting (16) in (20), we get the desired result.
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)
where \(T(f,g)\) is defined as in (5) and \(M,N,\) and \(k\) are as in Theorem 1.
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)
Substituting (12) in (22), we get the desired result.
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)
where \(T(f,g)\) is defined as in (5) and \(M,N\), and \(k\) are as in Theorem 1.
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)
Substituting (16) in (24), we get the desired result.
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.
