This study examines the multivariable Jensen integral inequality for convexity in coordinates. Using the co-ordinate convex functions, we give some results to develop new fractional Hermite-Hadamard type inequalities.
The inequalities discovered by Hermite and Hadamard for convex functions state that if \(f:I\subseteq \mathbb{R} \rightarrow \mathbb{R}\) is a convex mapping defined on the interval \(I\) of real numbers and \(a,b\in I\) with \(a<b\), then \[f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\int_{a}^{b}f(x) dx \leq \frac{f(a)+f(b)}{2}. \label{eq:HH} \tag{1}\]
Both inequalities hold in the reversed direction if \(f\) is concave [1].
Let \(0 < x_1 \le x_2 \le \dots \le x_n\) and let \(\lambda = (\lambda_1, \lambda_2, \dots, \lambda_n)\) be nonnegative weights such that \(\sum\limits_{j=1}^{n} \lambda_j = 1\). The well-known Jensen inequality [2] states that if \(f\) is convex on an interval containing \(x_n\), then \[f\Bigg( \sum\limits_{j=1}^{n} \lambda_j x_j \Bigg) \le \sum\limits_{j=1}^{n} \lambda_j f(x_j). \tag{2}\]
A central tool in the applied literature is Jensen’s weighted integral inequality. Assume that \(I\) is an arbitrary interval in \(\mathbb{R}\) and \(w,g:[a,b] \rightarrow I\) are integrable functions such that \(w>0\). If \(F:I \rightarrow \mathbb{R}\) is convex and \(F \circ g\) is integrable on \([a,b]\), then \[F\left( \frac{1}{W} \int_a^b w(x) g(x) dx \right) \le \frac{1}{W} \int_a^b w(x) F(g(x)) dx, \label{eq:JensenIntegral} \tag{3}\] where \[W = \int_a^b w(x) dx.\]
Inequality (3) is reversed if \(F\) is concave.
Inequalities like Hermite-Hadamard and Jensen play a central role in mathematical analysis, providing foundational tools in optimization, probability theory, and applied mathematics. The ongoing research in their extensions and applications underscores their relevance in modern mathematics.
The extension of classical inequalities to fractional integrals has been a prominent area of research. These studies aim to adapt convexity properties to fractional integral operators like the Riemann-Liouville and Hadamard fractional integrals. The fractional Jensen inequality explores how convexity is preserved under fractional integrals, often extending to generalized convex functions such as \(s\)-convex or \(h\)-convex functions. Similarly, the fractional Hermite-Hadamard inequality provides bounds for fractional integrals of convex and quasi-convex functions. These fractional inequalities have been further generalized to weighted fractional integrals and special classes of convex functions, such as harmonic and preinvex functions (see [1, 3– 9]).
Recall that \(f:\Delta = [a,b]\times [c,d] \rightarrow \mathbb{R}\) is convex on \(\Delta\) if \[f(tx+(1-t)y, tu+(1-t)v) \le t f(x,u) + (1-t) f(y,v), \quad \forall (x,u),(y,v) \in \Delta, \ t\in[0,1].\]
Definition 1 (Co-ordinated convex function).A function \(f:\Delta \rightarrow \mathbb{R}\) is called co-ordinated convex on \(\Delta\) if for all \((x,u),(y,v)\in \Delta\) and \(t,s \in [0,1]\), \[\begin{split} f(tx+(1-t)y, su+(1-s)v) \le ts f(x,u) + t(1-s) f(x,v) + s(1-t) f(y,u) + (1-t)(1-s) f(y,v). \end{split}\]
The function is co-ordinated concave if the inequality holds in the reversed direction.
A function \(f:\Delta \to \mathbb{R}\) is convex on the coordinates if the partial mappings \[f_y(u) = f(u,y) \text{ on } [a,b], \quad f_x(v) = f(x,v) \text{ on } [c,d],\] are convex for all \(x \in [a,b]\) and \(y \in [c,d]\) [10]. Every convex function is co-ordinated convex, but not vice versa.
Dragomir [10] proved the following Hermite-Hadamard type inequalities for co-ordinated convex functions on a rectangle:
Theorem 1 . Let \(f:\Delta \rightarrow \mathbb{R}\) be co-ordinated convex. Then \[f\Big(\frac{a+b}{2}, \frac{c+d}{2}\Big) \le \frac{1}{(b-a)(d-c)} \int_a^b \int_c^d f(x,y) \, dy \, dx \le \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}. \tag{5}\]
These inequalities are sharp. If \(f\) is co-ordinated concave, the inequalities reverse.
Definition 2(Riemann-Liouville fractional integral).Let \(f\in L_1[a,b]\). For \(\alpha>0\), define \[J_{a+}^\alpha f(x) = \frac{1}{\Gamma(\alpha)} \int_a^x (x-t)^{\alpha-1} f(t) dt, \quad x>a,\] \[J_{b-}^\alpha f(x) = \frac{1}{\Gamma(\alpha)} \int_x^b (t-x)^{\alpha-1} f(t) dt, \quad x<b.\]
Here \(\Gamma(x) = \int_0^\infty e^{-t} t^{x-1} dt\). Also, \(J_{a+}^0 f(x) = J_{b-}^0 f(x) = f(x)\).
Sarikaya et al. [6] proved the following fractional Hermite-Hadamard inequality:
Theorem 2. Let \(f:[a,b] \to [0,\infty)\) be convex and integrable on \([a,b]\) with \(a<b\). Then, for \(\alpha>0\), \[f\left(\frac{a+b}{2}\right) \le \frac{\Gamma(\alpha+1)}{2(b-a)^\alpha} \left[ J_{a+}^\alpha f(b) + J_{b-}^\alpha f(a) \right] \le \frac{f(a)+f(b)}{2}. \tag{6}\]
For two-variable functions, the Riemann-Liouville fractional integrals are defined as follows [11]:
Definition 3. Let \(f\in L_1([a,b]\times[c,d])\). Then \[J_{a+,c+}^{\alpha,\beta} f(x,y) = \frac{1}{\Gamma(\alpha)\Gamma(\beta)} \int_a^x \int_c^y (x-t)^{\alpha-1} (y-s)^{\beta-1} f(t,s) \, ds\, dt, \quad x>a, y>c,\] and similarly for \(J_{a+,d-}^{\alpha,\beta}, J_{b-,c+}^{\alpha,\beta}, J_{b-,d-}^{\alpha,\beta}\) as given in [11].
Sarikaya [11] also proved the following Hermite-Hadamard type inequality for two-variable co-ordinated convex functions:
Theorem 3. Let \(f:[a,b]\times[c,d] \to [0,\infty)\) be co-ordinated convex with \(0\le a<b\), \(0\le c<d\), and \(f \in L_1(\Delta)\). Then, for \(\alpha,\beta>0\), \[\begin{aligned} f\Big(\frac{a+b}{2}, \frac{c+d}{2}\Big) &\le \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4(b-a)^\alpha (d-c)^\beta} \Big[ J_{a+,c+}^{\alpha,\beta} f(b,d) + J_{a+,d-}^{\alpha,\beta} f(b,c) + J_{b-,c+}^{\alpha,\beta} f(a,d) + J_{b-,d-}^{\alpha,\beta} f(a,c) \Big] \\ &\le \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{4}. \end{aligned}\]
This study examines multivariable Jensen integral inequalities for convexity in coordinates and develops new fractional Hermite-Hadamard type inequalities generalizing the classical results.
To establish our main result, we recall the following standard facts (see, e.g., [12]).
A set \(I\subset \mathbb{R}^n\) is convex if \[t\mathbf{x}+(1-t)\mathbf{y}\in I \quad \text{for all } \mathbf{x},\mathbf{y} \in I \text{ and } t\in[0,1].\]
A function \(F:I\to \mathbb{R}\) is convex if \[F\big(t\mathbf{x}+(1-t)\mathbf{y}\big)\le tF(\mathbf{x})+(1-t)F(\mathbf{y}) \quad \text{for all } \mathbf{x},\mathbf{y}\in I,\ t\in[0,1].\]
We also say that \(F\) admits a supporting hyperplane at \(\mathbf{y}\in I\) if there exists \(\boldsymbol{\lambda}\in \mathbb{R}^n\) such that \[F(\mathbf{x})-F(\mathbf{y})\ge \sum\limits_{k=1}^n \lambda_k\,(x_k-y_k) \quad \text{ for all } \mathbf{x}\in I .\]
Lemma 1. Let \(I\subset \mathbb{R}^n\) be a convex set and \(F:I\to \mathbb{R}\). Then the following are equivalent:
1. \(F\) is convex on \(I\).
2. \(F\) is differentiable on \(I\) and, for each \(\mathbf{y}\in I\), \[F(\mathbf{x})-F(\mathbf{y})\ \geq \ \nabla F(\mathbf{y})^{\! \top }(\mathbf{x }-\mathbf{y})\quad \text{for all }\mathbf{x}\in I.\]
If, in addition, \(I\) has nonempty interior and \(F\in C^{2}(I)\), then the above are also equivalent to:
3. The Hessian matrix \[\nabla ^{2}F(\mathbf{y})=\bigg(\frac{\partial ^{2}F}{\partial x_{i}\, \partial x_{j}}(\mathbf{y})\bigg)_{1\leq i,j\leq n},\] is positive semidefinite for every \(\mathbf{y}\in I\).
Moreover, if any (hence all) of the above hold, then at every \(\mathbf{y}\in I\) there exists a vector \(\boldsymbol{\lambda}\in \mathbb{R}^{n}\) (e.g., \(\boldsymbol{\lambda}=\nabla F(\mathbf{y})\) when \(F\) is differentiable) such that \[F(\mathbf{x})-F(\mathbf{y})\ \ge \ \sum\limits_{k=1}^{n}\lambda_k(x_k-y_k)\quad \text{for all }\mathbf{x}\in I,\] i.e., \(F\) admits a supporting hyperplane at every point of \(I\).
Theorem 4 (Jensen inequality with multiple variables). Suppose that \(F:\Delta =[a,b]\times \lbrack c,d]\rightarrow \mathbb{R}\) is continuous and coordinate-wise convex on \(\Delta\). Let \(h_{1},g_{1}:[a,b]\rightarrow \mathbb{R}\) and \(h_{2},g_{2}:[c,d]\rightarrow \mathbb{R}\) be integrable functions with \(h_{1},h_{2}>0\). Define \[H_{1}=\int_{a}^{b}h_{1}(x)\,dx,\quad H_{2}=\int_{c}^{d}h_{2}(y)\,dy.\]
Then, we have \[F\Bigg(\frac{1}{H_{1}}\int_{a}^{b}h_{1}(x)g_{1}(x)\,dx,\ \frac{1}{H_{2}} \int_{c}^{d}h_{2}(y)g_{2}(y)\,dy\Bigg)\leq \frac{1}{H_{1}H_{2}} \int_{a}^{b}\int_{c}^{d}h_{1}(x)h_{2}(y)F(g_{1}(x),g_{2}(y))\,dy\,dx.\]
Proof. Let \[(x_{1},y_{1})=\Bigg(\frac{1}{H_{1}}\int_{a}^{b}h_{1}(x)g_{1}(x)\,dx,\ \frac{1 }{H_{2}}\int_{c}^{d}h_{2}(y)g_{2}(y)\,dy\Bigg)\in \Delta .\]
Since \(F\) is convex and differentiable, from Lemma 1 (3) there exists \[\nabla F(x_{1},y_{1})=\left( \frac{\partial F}{\partial x}(x_{1},y_{1}), \frac{\partial F}{\partial y}(x_{1},y_{1})\right),\] such that \[F(x,y)-F(x_{1},y_{1})\geq \frac{\partial F}{\partial x} (x_{1},y_{1})(x-x_{1})+\frac{\partial F}{\partial y}(x_{1},y_{1})(y-y_{1}), \label{eq:grad} \tag{7}\] for all \((x,y)\in \Delta\).
Multiplying both sides of (7) by \(h_{1}(x)h_{2}(y)\) and integrating over \([a,b]\times \lbrack c,d]\), we obtain \[\begin{aligned} \int_{a}^{b}\int_{c}^{d}&h_{1}(x)h_{2}(y)\big[ F(g_{1}(x),g_{2}(y))-F(x_{1},y_{1})\big]\,dy\,dx \\ & \geq \int_{a}^{b}\int_{c}^{d}h_{1}(x)h_{2}(y)\left[ \frac{\partial F}{ \partial x}(x_{1},y_{1})(g_{1}(x)-x_{1})+\frac{\partial F}{\partial y} (x_{1},y_{1})(g_{2}(y)-y_{1})\right] dy\,dx \\ & =\frac{\partial F}{\partial x}(x_{1},y_{1})\underbrace{ \int_{a}^{b}h_{1}(x)(g_{1}(x)-x_{1})dx}_{=0}\underbrace{ \int_{c}^{d}h_{2}(y)dy}_{=H_{2}}+\frac{\partial F}{\partial y}(x_{1},y_{1}) \underbrace{\int_{c}^{d}h_{2}(y)(g_{2}(y)-y_{1})dy}_{=0}\underbrace{ \int_{a}^{b}h_{1}(x)dx}_{=H_{1}} \\ & =0. \end{aligned}\]
Dividing both sides by \(H_{1}H_{2}\), we obtain the desired inequality: \[F(x_{1},y_{1})\leq \frac{1}{H_{1}H_{2}}\int_{a}^{b} \int_{c}^{d}h_{1}(x)h_{2}(y)F(g_{1}(x),g_{2}(y))\,dy\,dx.\] ◻
Remark 1. Theorem 4 states a generalization of the classical Jensen inequality to functions of two variables that are convex in each coordinate.
In particular, if we take \(h_1(x) = H_1/(b-a)\) and \(h_2(y) = H_2/(d-c)\) as constant functions, then the weighted averages reduce to the standard arithmetic averages: \[\frac{1}{H_1} \int_a^b h_1(x) g_1(x)\,dx = \frac{1}{b-a} \int_a^b g_1(x)\,dx, \quad \frac{1}{H_2} \int_c^d h_2(y) g_2(y)\,dy = \frac{1}{d-c} \int_c^d g_2(y)\,dy.\]
Consequently, Theorem 4 recovers the classical two-variable Jensen inequality for coordinate-wise convex functions: \[F\Bigg(\frac{1}{b-a}\int_{a}^{b}g_{1}(x)\,dx,\frac{1}{d-c} \int_{c}^{d}g_{2}(y)\,dy\Bigg)\leq \frac{1}{(b-a)(d-c)}\int_{a}^{b} \int_{c}^{d}F(g_{1}(x),g_{2}(y))\,dy\,dx.\]
Theorem 5. Let \(f:\Delta =[a,b]\times \lbrack c,d]\rightarrow \mathbb{R}\) be a co-ordinated convex function on \(\Delta\). Then, for \(\alpha ,\beta >0\), the following inequalities hold: \[\begin{aligned} \label{1} &f\Big(\frac{b+\alpha a}{\alpha +1},\frac{d+\beta c}{\beta +1}\Big)+f\Big( \frac{b+\alpha a}{\alpha +1},\frac{\beta d+c}{\beta +1}\Big) +f\Big(\frac{\alpha b+a}{\alpha +1},\frac{d+\beta c}{\beta +1}\Big)+f\Big( \frac{\alpha b+a}{\alpha +1},\frac{\beta d+c}{\beta +1}\Big) \notag \\ &\qquad \leq \frac{\Gamma (\alpha +1)\Gamma (\beta +1)}{(b-a)^{\alpha }(d-c)^{\beta }}\Big[J_{a+,c+}^{\alpha ,\beta }f(b,d)+J_{a+,d-}^{\alpha ,\beta }f(b,c) +J_{b-,c+}^{\alpha ,\beta }f(a,d)+J_{b-,d-}^{\alpha ,\beta }f(a,c) \Big] \notag \\ &\qquad \leq f(a,c)+f(a,d)+f(b,c)+f(b,d). \end{aligned} \tag{8}\]
Proof. By the Jensen inequality for weighted integrals with multiple variables, we have \[\begin{aligned} & f\Bigg(\frac{\alpha }{(b-a)^{\alpha }}\int_{a}^{b}(b-x)^{\alpha -1}x\,dx,\ \frac{\beta }{(d-c)^{\beta }}\int_{c}^{d}(d-y)^{\beta -1}y\,dy\Bigg) \\ &\qquad \leq \frac{\alpha \beta }{(b-a)^{\alpha }(d-c)^{\beta }} \int_{a}^{b}\int_{c}^{d}(b-x)^{\alpha -1}(d-y)^{\beta -1}f(x,y)\,dy\,dx, \end{aligned}\] and similarly for the three other combinations: \[(x-a)^{\alpha -1},\ (b-x)^{\alpha -1}\quad \text{and}\quad (y-c)^{\beta -1},\ (d-y)^{\beta -1}.\]
Adding these four inequalities, we obtain \[\begin{aligned} & f\Big(\frac{b+\alpha a}{\alpha +1},\frac{d+\beta c}{\beta +1}\Big)+f\Big( \frac{b+\alpha a}{\alpha +1},\frac{\beta d+c}{\beta +1}\Big) +f\Big(\frac{\alpha b+a}{\alpha +1},\frac{d+\beta c}{\beta +1}\Big)+f\Big( \frac{\alpha b+a}{\alpha +1},\frac{\beta d+c}{\beta +1}\Big) \notag \\ & \quad \leq \frac{\alpha \beta }{(b-a)^{\alpha }(d-c)^{\beta }}\Bigg[ \int_{a}^{b}\int_{c}^{d}(b-x)^{\alpha -1}(d-y)^{\beta -1}f(x,y)\,dy\,dx \notag \\ & \qquad +\int_{a}^{b}\int_{c}^{d}(b-x)^{\alpha -1}(y-c)^{\beta -1}f(x,y)\,dy\,dx \quad +\int_{a}^{b}\int_{c}^{d}(x-a)^{\alpha -1}(d-y)^{\beta -1}f(x,y)\,dy\,dx \notag \\ & \qquad +\int_{a}^{b}\int_{c}^{d}(x-a)^{\alpha -1}(y-c)^{\beta -1}f(x,y)\,dy\,dx\Bigg]. \notag \end{aligned}\]
Now, applying the change of variables \[x=ta+(1-t)b,\quad dx=(b-a)dt,\quad y=sc+(1-s)d,\quad dy=(d-c)ds,\quad t,s\in \lbrack 0,1],\] and using the co-ordinated convexity of \(f\), we obtain \[\begin{aligned} & f\Big(\frac{b+\alpha a}{\alpha +1},\frac{d+\beta c}{\beta +1}\Big)+f\Big( \frac{b+\alpha a}{\alpha +1},\frac{\beta d+c}{\beta +1}\Big) +f\Big(\frac{\alpha b+a}{\alpha +1},\frac{d+\beta c}{\beta +1}\Big)+f\Big( \frac{\alpha b+a}{\alpha +1},\frac{\beta d+c}{\beta +1}\Big) \\ &\quad \leq \frac{\Gamma (\alpha +1)\Gamma (\beta +1)}{(b-a)^{\alpha }(d-c)^{\beta }}\Big[J_{a+,c+}^{\alpha ,\beta }f(b,d)+J_{a+,d-}^{\alpha ,\beta }f(b,c) +J_{b-,c+}^{\alpha ,\beta }f(a,d)+J_{b-,d-}^{\alpha ,\beta }f(a,c) \Big]\\ &\quad \leq f(a,c)+f(a,d)+f(b,c)+f(b,d), \end{aligned}\] which completes the proof. ◻
Remark 2. Under the assumptions of Theorem 5, if we take \(\alpha =\beta =1\), then the fractional integral inequality (8) reduces to the classical Hermite–Hadamard inequality for functions that are convex on each coordinate.
Indeed, for \(\alpha =\beta =1\), the Riemann–Liouville fractional integrals become the standard double integrals, and (8) simplifies to \[f\left( \frac{a+b}{2},\frac{c+d}{2}\right) \leq \frac{1}{(b-a)(d-c)} \int_{a}^{b}\int_{c}^{d}f(x,y)\,dy\,dx\leq \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d) }{4}.\]
Thus, Theorem 5 can be seen as a generalization of the classical Hermite–Hadamard inequality to the setting of Riemann–Liouville fractional integrals with arbitrary positive parameters \(\alpha\) and \(\beta\).
Similarly, we obtain the following generalized Hermite-Hadamard inequalities for fractional operator:
Theorem 6. Let \(\Delta=[a,b]\times[c,d]\) with \(0\le a<b\) and \(0\le c<d\), fix \(n\in \mathbb{N}\), and let \(\alpha,\beta>0\). Assume \(f:\Delta \to \mathbb{R}\) is co-ordinated convex and \[(t,s)\mapsto f(t^{n},s^{n})\in L^{1}(\Delta).\]
(Equivalently, the assumption on \(a,b,c,d\ge0\) may be dropped if \(n\) is even.) Then the following inequalities hold: \[\begin{aligned} \label{2} &f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( b-x\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( d-y\right) ^{\beta -1}\left( y^{n}\right) dy\right) \notag\\ &\qquad+f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( b-x\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( y-c\right) ^{\beta -1}\left( y^{n}\right) dy\right) \notag \\ &\qquad+f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( x-a\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( d-y\right) ^{\beta -1}\left( y^{n}\right) dy\right) \notag \\ &\qquad+f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( x-a\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( y-c\right) ^{\beta -1}\left( y^{n}\right) dy\right) \notag \\ &\quad\leq \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{ \left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\left[ J_{a+,c+}^{\alpha ,\beta }f(b^{n},d^{n})+J_{a+,d-}^{\alpha ,\beta }f(b^{n},c^{n})+J_{b-,c+}^{\alpha ,\beta }f(a^{n},b^{n})+J_{b-,d-}^{\alpha ,\beta }f(a^{n},c^{n})\right] \notag \\ &\quad\leq \alpha \beta \sum \limits_{k_{1}=0}^{n}\sum \limits_{k_{2}=0}^{n} \left[ \frac{\left( \alpha +n-k_{1}-1\right) !n!}{\left( n-k_{1}\right) !\left( \alpha +n\right) !}+\frac{\left( \alpha +k_{2}-1\right) !n!}{ k_{1}!\left( \alpha +n\right) !}\right] \notag \\ &\qquad\times \left[ \frac{\left( \beta +n-k_{2}-1\right) !n!}{\left( n-k_{2}\right) !\left( \beta +n\right) !}+\frac{\left( \beta +k_{2}-1\right) !n!}{k_{2}!\left( \beta +n\right) !}\right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) . \end{aligned} \tag{9}\]
Proof. Using Jensen integral inequality with multible variables, we have \[\begin{aligned} &f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( b-x\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( d-y\right) ^{\beta -1}\left( y^{n}\right) dy\right) \\ &\qquad \leq \frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f\left( x^{n},y^{n}\right) dydx,\\ &f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( b-x\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( y-c\right) ^{\beta -1}\left( y^{n}\right) dy\right) \\ &\quad\leq \frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f\left( x^{n},y^{n}\right) dydx\\ &f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( x-a\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( d-y\right) ^{\beta -1}\left( y^{n}\right) dy\right) \\ &\quad\leq \frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f\left( x^{n},y^{n}\right) dydx, \end{aligned}\] and \[\begin{aligned} &f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( x-a\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( y-c\right) ^{\beta -1}\left( y^{n}\right) dy\right) \\ &\quad\leq \frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f\left( x^{n},y^{n}\right) dydx, \end{aligned}\] for all \(\left( x,y\right) \in \Delta .\) By adding these inequalities, we get \[\begin{aligned} \label{q2} &f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( b-x\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( d-y\right) ^{\beta -1}\left( y^{n}\right) dy\right) \notag \\ &\qquad+f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( b-x\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( y-c\right) ^{\beta -1}\left( y^{n}\right) dy\right) \notag \\ &\qquad+f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( x-a\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( d-y\right) ^{\beta -1}\left( y^{n}\right) dy\right) \notag \\ &\qquad+f\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }}\int_{a}^{b}\left( x-a\right) ^{\alpha -1}\left( x^{n}\right) dx,\frac{\beta }{\left( d-c\right) ^{\beta }}\int_{c}^{d}\left( y-c\right) ^{\beta -1}\left( y^{n}\right) dy\right) \notag \\ &\quad\leq \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{ \left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\left[ J_{a+,c+}^{\alpha ,\beta }f(b^{n},d^{n})+J_{a+,d-}^{\alpha ,\beta }f(b^{n},c^{n})+J_{b-,c+}^{\alpha ,\beta }f(a^{n},b^{n})+J_{b-,d-}^{\alpha ,\beta }f(a^{n},c^{n})\right] . \end{aligned} \tag{10}\]
By changing of the variables \(x=ta+\left( 1-t\right) b\), \(y=sc+\left( 1-s\right) d\) and \(x=\left( 1-t\right) a+tb,\) \(y=\left( 1-s\right) c+sd\) for \(\left( x,y\right) \in \Delta\) and \(\left( t,s\right) \in \left[ 0,1\right] ^{2}\) in (10) respectively\(,\) we obtain \[\begin{aligned} &f\left( \alpha \int_{0}^{1}t^{\alpha -1}\left( ta+\left( 1-t\right) b\right) ^{n}dt,\beta \int_{0}^{1}s^{\beta -1}\left( sc+\left( 1-s\right) d\right) ^{n}ds\right) \\ &\qquad+f\left( \alpha \int_{0}^{1}t^{\alpha -1}\left( ta+\left( 1-t\right) b\right) ^{n}dt,\beta \int_{0}^{1}s^{\beta -1}\left( \left( 1-s\right) c+sd\right) ^{n}ds\right) \\ &\qquad+f\left( \alpha \int_{0}^{1}t^{\alpha -1}\left( \left( 1-t\right) a+tb\right) ^{n}dt,\beta \int_{0}^{1}s^{\beta -1}\left( sc+\left( 1-s\right) d\right) ^{n}ds\right) \\ &\qquad+f\left( \alpha \int_{0}^{1}t^{\alpha -1}\left( \left( 1-t\right) a+tb\right) ^{n}dt,\beta \int_{0}^{1}s^{\beta -1}\left( \left( 1-s\right) c+sd\right) ^{n}ds\right) \\ &\quad\leq \alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \left( ta+\left( 1-t\right) b\right) ^{n},\left( sc+\left( 1-s\right) d\right) ^{n}\right] dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \left( ta+\left( 1-t\right) b\right) ^{n},\left( \left( 1-s\right) c+sd\right) ^{n}\right] dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \left( \left( 1-t\right) a+tb\right) ^{n},\left( sc+\left( 1-s\right) d\right) ^{n}\right] dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \left( \left( 1-t\right) a+tb\right) ^{n},\left( \left( 1-s\right) c+sd\right) ^{n}\right] dsdt, \end{aligned}\] and so \[\begin{aligned} &f\left( \alpha \int_{0}^{1}t^{\alpha -1}\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) t^{n-k}a^{n-k}\left( 1-t\right) ^{k}b^{k}dt,\beta \int_{0}^{1}s^{\beta -1}\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) s^{n-k}c^{n-k}\left( 1-s\right) ^{k}d^{k}ds\right) \\ &\qquad+f\left( \alpha \int_{0}^{1}t^{\alpha -1}\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) t^{n-k}a^{n-k}\left( 1-t\right) ^{k}b^{k}dt,\beta \int_{0}^{1}s^{\beta -1}\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-s\right) ^{n-k}c^{n-k}s^{k}d^{k}ds\right) \\ &\qquad+f\left( \alpha \int_{0}^{1}t^{\alpha -1}\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-t\right) ^{n-k}a^{n-k}t^{k}b^{k}dt,\beta \int_{0}^{1}s^{\beta -1}\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) s^{n-k}c^{n-k}\left( 1-s\right) ^{k}d^{k}ds\right) \\ &\qquad+f\left( \alpha \int_{0}^{1}t^{\alpha -1}\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-t\right) ^{n-k}a^{n-k}t^{k}b^{k}dt,\beta \int_{0}^{1}s^{\beta -1}\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-s\right) ^{n-k}c^{n-k}s^{k}d^{k}ds\right) \\ &\quad\leq \alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) t^{n-k}a^{n-k}\left( 1-t\right) ^{k}b^{k},\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) s^{n-k}c^{n-k}\left( 1-s\right) ^{k}d^{k}\right] dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) t^{n-k}a^{n-k}\left( 1-t\right) ^{k}b^{k},\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-s\right) ^{n-k}c^{n-k}s^{k}d^{k}\right] dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-t\right) ^{n-k}a^{n-k}t^{k}b^{k},\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) s^{n-k}c^{n-k}\left( 1-s\right) ^{k}d^{k}\right] dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-t\right) ^{n-k}a^{n-k}t^{k}b^{k},\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-s\right) ^{n-k}c^{n-k}s^{k}d^{k}\right] dsdt. \end{aligned}\]
Since \(\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) t^{k}\left( 1-t\right) ^{n-k}=1\) and \(\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) s^{k}\left( 1-s\right) ^{n-k}=1,\) by the co-ordinated convexity of \(f\), and considering \[\int_{0}^{1}t^{m-1}\left( 1-t\right) ^{n-1}dt=B\left( m,n\right) =\frac{ \left( m-1\right) !\left( n-1\right) !}{\left( m+n-1\right) !},\] it follows that \[\begin{aligned} &f\left( \alpha \int_{0}^{1}\sum \limits_{k=0}^{n}\frac{n!\left( \alpha +n-k-1\right) !}{(n-k)!\left( \alpha +n\right) !}a^{n-k}b^{k}dt,\beta \int_{0}^{1}\sum \limits_{k=0}^{n}\frac{n!\left( \beta +n-k-1\right) !}{ (n-k)!\left( \beta +n\right) !}c^{n-k}d^{k}ds\right) \\ &\qquad+f\left( \alpha \int_{0}^{1}\sum \limits_{k=0}^{n}\frac{n!\left( \alpha +n-k-1\right) !}{(n-k)!\left( \alpha +n\right) !}a^{n-k}b^{k}dt,\beta \int_{0}^{1}\sum \limits_{k=0}^{n}\frac{n!\left( \beta +k-1\right) !}{ k!\left( \beta +n\right) !}c^{n-k}d^{k}ds\right) \\ &\qquad+f\left( \alpha \int_{0}^{1}\sum \limits_{k=0}^{n}\frac{n!\left( \alpha +k-1\right) !}{k!\left( \alpha +n\right) !}a^{n-k}b^{k}dt,\beta \int_{0}^{1}\sum \limits_{k=0}^{n}\frac{n!\left( \beta +n-k-1\right) !}{ (n-k)!\left( \beta +n\right) !}c^{n-k}d^{k}ds\right) \\ &\qquad+f\left( \alpha \int_{0}^{1}\sum \limits_{k=0}^{n}\frac{n!\left( \alpha +k-1\right) !}{k!\left( \alpha +n\right) !}a^{n-k}b^{k}dt,\beta \int_{0}^{1}\sum \limits_{k=0}^{n}\frac{n!\left( \beta +k-1\right) !}{ k!\left( \beta +n\right) !}c^{n-k}d^{k}ds\right) \\ &\quad\leq \alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) t^{n-k}\left( 1-t\right) ^{k}a^{n-k}b^{k},\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) s^{n-k}\left( 1-s\right) ^{k}c^{n-k}d^{k}\right] dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) t^{n-k}\left( 1-t\right) ^{k}a^{n-k}b^{k},\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) s^{k}\left( 1-s\right) ^{n-k}c^{n-k}d^{k}\right] dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-t\right) ^{n-k}t^{k}a^{n-k}b^{k},\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) s^{n-k}\left( 1-s\right) ^{k}c^{n-k}d^{k}\right] dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f\left[ \sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) \left( 1-t\right) ^{n-k}t^{k}a^{n-k}b^{k},\sum \limits_{k=0}^{n}\left( \begin{array}{c} n \\ k \end{array} \right) s^{k}\left( 1-s\right) ^{n-k}c^{n-k}d^{k}\right] dsdt \\ &\quad\leq \alpha \beta \int_{0}^{1}\int_{0}^{1}\left( \sum \limits_{k_{1}=0}^{n}\left( \begin{array}{c} n \\ k_{1} \end{array} \right) t^{\alpha +n-k_{1}-1}\left( 1-t\right) ^{k_{1}}\right) \\ &\qquad\times \left( \sum \limits_{k_{2}=0}^{n}\left( \begin{array}{c} n \\ k_{2} \end{array} \right) s^{\beta +n-k_{2}-1}\left( 1-s\right) ^{k_{2}}\right) f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}\left( \sum \limits_{k_{1}=0}^{n}\left( \begin{array}{c} n \\ k_{1} \end{array} \right) t^{\alpha +n-k_{1}-1}\left( 1-t\right) ^{k_{1}}\right) \\ &\qquad\times \left( \sum \limits_{k_{2}=0}^{n}\left( \begin{array}{c} n \\ k_{2} \end{array} \right) s^{\beta +k_{2}-1}\left( 1-s\right) ^{n-k_{2}}\right) f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}\left( \sum \limits_{k_{1}=0}^{n}\left( \begin{array}{c} n \\ k_{1} \end{array} \right) t^{\alpha +k_{1}-1}\left( 1-t\right) ^{n-k_{1}}\right) \\ &\qquad\times \left( \sum \limits_{k_{2}=0}^{n}\left( \begin{array}{c} n \\ k_{2} \end{array} \right) s^{\beta +n-k_{2}-1}\left( 1-s\right) ^{k_{2}}\right) f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}\left( \sum \limits_{k_{1}=0}^{n}\left( \begin{array}{c} n \\ k_{1} \end{array} \right) t^{\alpha +k_{1}-1}\left( 1-t\right) ^{n-k_{1}}\right) \\ &\qquad\times \left( \sum \limits_{k_{2}=0}^{n}\left( \begin{array}{c} n \\ k_{2} \end{array} \right) s^{\beta +k_{2}-1}\left( 1-s\right) ^{n-k_{2}}\right) f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) dsdt \\ &\quad=\alpha \beta \left[ \sum \limits_{k_{1}=0}^{n}\left( \begin{array}{c} n \\ k_{1} \end{array} \right) \int_{0}^{1}\left( t^{\alpha +n-k_{1}-1}\left( 1-t\right) ^{k_{1}}\right) dt\right] \\ &\qquad\times \left[ \sum \limits_{k_{2}=0}^{n}\left( \begin{array}{c} n \\ k_{2} \end{array} \right) \int_{0}^{1}\left( s^{\beta +n-k_{2}-1}\left( 1-s\right) ^{k_{2}}\right) ds\right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) \\ &\qquad+\alpha \beta \left[ \sum \limits_{k_{1}=0}^{n}\left( \begin{array}{c} n \\ k_{1} \end{array} \right) \int_{0}^{1}\left( t^{\alpha +n-k_{1}-1}\left( 1-t\right) ^{k_{1}}\right) dt\right] \\ &\qquad\times \left[ \sum \limits_{k_{2}=0}^{n}\left( \begin{array}{c} n \\ k_{2} \end{array} \right) \int_{0}^{1}\left( s^{\beta +k_{2}-1}\left( 1-s\right) ^{n-k_{2}}\right) ds\right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) \\ &\qquad+\alpha \beta \left[ \sum \limits_{k_{1}=0}^{n}\left( \begin{array}{c} n \\ k_{1} \end{array} \right) \int_{0}^{1}t^{\alpha +k_{1}-1}\left( 1-t\right) ^{n-k_{1}}dt\right] \\ &\qquad\times \left[ \sum \limits_{k_{2}=0}^{n}\left( \begin{array}{c} n \\ k_{2} \end{array} \right) \int_{0}^{1}s^{\beta +n-k_{2}-1}\left( 1-s\right) ^{k_{2}}ds\right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) \\ &\qquad+\alpha \beta \left[ \sum \limits_{k_{1}=0}^{n}\left( \begin{array}{c} n \\ k_{1} \end{array} \right) \int_{0}^{1}t^{\alpha +k_{1}-1}\left( 1-t\right) ^{n-k_{1}}dt\right] \\ &\qquad\times \left[ \sum \limits_{k_{2}=0}^{n}\left( \begin{array}{c} n \\ k_{2} \end{array} \right) \int_{0}^{1}s^{\beta +k_{2}-1}\left( 1-s\right) ^{n-k_{2}}ds\right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) \\ &\quad=\alpha \beta \left[ \sum \limits_{k_{1}=0}^{n}\frac{\left( \alpha +n-k_{1}-1\right) !n!}{\left( n-k_{1}\right) !\left( \alpha +n\right) !} \right] \left[ \sum \limits_{k_{2}=0}^{n}\frac{\left( \beta +n-k_{2}-1\right) !n!}{\left( n-k_{2}\right) !\left( \beta +n\right) !} \right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) \\ &\qquad+\alpha \beta \left[ \sum \limits_{k_{1}=0}^{n}\frac{\left( \alpha +n-k_{1}-1\right) !n!}{\left( n-k_{1}\right) !\left( \alpha +n\right) !} \right] \left[ \sum \limits_{k_{2}=0}^{n}\frac{\left( \beta +k_{2}-1\right) !n!}{k_{2}!\left( \beta +n\right) !}\right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) \\ &\qquad+\alpha \beta \left[ \sum \limits_{k_{1}=0}^{n}\frac{\left( \alpha +k_{2}-1\right) !n!}{k_{1}!\left( \alpha +n\right) !}\right] \left[ \sum \limits_{k_{2}=0}^{n}\frac{\left( \beta +n-k_{2}-1\right) !n!}{\left( n-k_{2}\right) !\left( \beta +n\right) !}\right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) \\ &\qquad+\alpha \beta \left[ \sum \limits_{k_{1}=0}^{n}\frac{\left( \alpha +k_{2}-1\right) !n!}{k_{1}!\left( \alpha +n\right) !}\right] \left[ \sum \limits_{k_{2}=0}^{n}\frac{\left( \beta +k_{2}-1\right) !n!}{k_{2}!\left( \beta +n\right) !}\right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) \\ &\quad=\alpha \beta \sum \limits_{k_{1}=0}^{n}\sum \limits_{k_{2}=0}^{n}\left[ \frac{\left( \alpha +n-k_{1}-1\right) !n!}{\left( n-k_{1}\right) !\left( \alpha +n\right) !}+\frac{\left( \alpha +k_{2}-1\right) !n!}{k_{1}!\left( \alpha +n\right) !}\right] \\ &\qquad\times \left[ \frac{\left( \beta +n-k_{2}-1\right) !n!}{\left( n-k_{2}\right) !\left( \beta +n\right) !}+\frac{\left( \beta +k_{2}-1\right) !n!}{k_{2}!\left( \beta +n\right) !}\right] f\left( a^{n-k_{1}}b^{k_{1}},c^{n-k_{2}}d^{k_{2}}\right) \end{aligned}\] which is completed the inequalities of (9). ◻
Remark 3. Under the assumptions of Theorem 6 , if we take n = 1, then the inequality (9) reduces to the fractional integral inequality (8). Indeed, for n = 1, we have xn = x and yn = y, so all terms in (9) involving xn and yn become the standard terms in (8).
Moreover, when n > 1, the requirement that f is positive or that n is even ensures that the expressions xn and yn are well-defined over the domain Δ even if a, b, c, or d are negative. Hence, Theorem 6 generalizes (8) to powers n of the variables while maintaining the validity of the inequality under these conditions.
Now, we give the new following result.
Theorem 7. Let \(f:\Delta \rightarrow \mathbb{R}\) be a co-ordinated convex function on \(\Delta\). Let \(n \in \mathbb{N}\), and assume either \(f(x,y) \ge 0\) for all \((x,y) \in \Delta\) or \(n\) is an even integer. Then the following inequalities hold for \(\alpha, \beta > 0\): \[\begin{aligned} \label{3} &f^{n}\left( \frac{b+\alpha a}{\alpha +1},\frac{d+\beta c}{\beta +1}\right) +f^{n}\left( \frac{b+\alpha a}{\alpha +1},\frac{\beta d+c}{\beta +1}\right) +f^{n}\left( \frac{\alpha b+a}{\alpha +1},\frac{d+\beta c}{\beta +1} \right) +f^{n}\left( \frac{\alpha b+a}{\alpha +1},\frac{\beta d+c}{\beta +1} \right) \notag \\ &\quad\leq \left[ \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }} J_{a+,c+}^{\alpha ,\beta }f(b,d)\right] ^{n}+\left[ \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}J_{a+,d-}^{\alpha ,\beta }f(b,c)\right] ^{n} \notag \\ &\qquad+\left[ \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}J_{b-,c+}^{\alpha ,\beta }f(a,b)\right] ^{n}+\left[ \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}J_{b-,d-}^{\alpha ,\beta }f(a,c)\right] ^{n} \notag \\ &\quad\leq \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{ \left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\left[ J_{a+,c+}^{\alpha ,\beta }f^{n}(b,d)+J_{a+,d-}^{\alpha ,\beta }f^{n}(b,c)+J_{b-,c+}^{\alpha ,\beta }f^{n}(a,b)+J_{b-,d-}^{\alpha ,\beta }f^{n}(a,c)\right] \notag \\ &\quad\leq \alpha \beta \sum \limits_{k_{1}+k_{2}+k_{3}+k_{4}=n}\frac{n!}{ k_{1}!k_{2}!k_{3}!k_{4}!}\left[ \frac{\left( \alpha +k_{1}+k_{2}-1\right) !\left( k_{3}+k_{4}\right) !}{\left( \alpha +n\right) !}+\frac{\left( \alpha +k_{3}+k_{4}-1\right) !\left( k_{1}+k_{2}\right) !}{\left( \alpha +n\right) ! }\right] \notag \\ &\qquad\times \left[ \frac{\left( \beta +k_{1}+k_{3}-1\right) !\left( k_{2}+k_{4}\right) !}{\left( \alpha +n\right) !}+\frac{\left( \beta +k_{2}+k_{4}-1\right) !\left( k_{1}+k_{3}\right) !}{\left( \alpha +n\right) ! }\right] \times f^{k_{1}}\left( a,c\right) f^{k_{2}}\left( a,d\right) f^{k_{3}}\left( b,c\right) f^{k_{4}}\left( b,d\right) . \end{aligned} \tag{11}\]
Proof. By the co-ordinated convexity of \(f\), \(F(x)=x^{n}\) and by Jensen integral inequality with multible variables, we get \[\begin{aligned} f^{n}\left( \frac{b+\alpha a}{\alpha +1},\frac{d+\beta c}{\beta +1}\right) &=&f^{n}\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }} \int_{a}^{b}\left( b-x\right) ^{\alpha -1}\left( x\right) dx,\frac{\beta }{ \left( d-c\right) ^{\beta }}\int_{c}^{d}\left( d-y\right) ^{\beta -1}\left( y\right) dy\right) \\ &\leq &\left[ \frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f\left( x,y\right) dydx\right] ^{n} \\ &\leq &\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f^{n}\left( x,y\right) dydx,\\ f^{n}\left( \frac{b+\alpha a}{\alpha +1},\frac{\beta d+c}{\beta +1}\right) &=&f^{n}\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }} \int_{a}^{b}\left( b-x\right) ^{\alpha -1}\left( x\right) dx,\frac{\beta }{ \left( d-c\right) ^{\beta }}\int_{c}^{d}\left( y-c\right) ^{\beta -1}\left( y\right) dy\right) \\ &\leq &\left[ \frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f\left( x,y\right) dydx\right] ^{n} \\ &\leq &\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f^{n}\left( x,y\right) dydx,\\ f^{n}\left( \frac{\alpha b+a}{\alpha +1},\frac{d+\beta c}{\beta +1}\right) &=&f^{n}\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }} \int_{a}^{b}\left( x-a\right) ^{\alpha -1}\left( x\right) dx,\frac{\beta }{ \left( d-c\right) ^{\beta }}\int_{c}^{d}\left( d-y\right) ^{\beta -1}\left( y\right) dy\right) \\ &\leq &\left[ \frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f\left( x,y\right) dydx\right] ^{n} \\ &\leq &\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f^{n}\left( x,y\right) dydx, \end{aligned}\] and \[\begin{aligned} f^{n}\left( \frac{\alpha b+a}{\alpha +1},\frac{\beta d+c}{\beta +1}\right) &=&f^{n}\left( \frac{\alpha }{\left( b-a\right) ^{\alpha }} \int_{a}^{b}\left( x-a\right) ^{\alpha -1}\left( x\right) dx,\frac{\beta }{ \left( d-c\right) ^{\beta }}\int_{c}^{d}\left( y-c\right) ^{\beta -1}\left( y\right) dy\right) \\ &\leq &\left[ \frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f\left( x,y\right) dydx\right] ^{n} \\ &\leq &\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f^{n}\left( x,y\right) dydx, \end{aligned}\] for all \(\left( x,y\right) \in \Delta .\) Adding both sides of above results, we can write \[\begin{aligned} \label{q5} &f^{n}\left( \frac{b+\alpha a}{\alpha +1},\frac{d+\beta c}{\beta +1}\right) +f^{n}\left( \frac{b+\alpha a}{\alpha +1},\frac{\beta d+c}{\beta +1}\right) +f^{n}\left( \frac{\alpha b+a}{\alpha +1},\frac{d+\beta c}{\beta +1} \right) +f^{n}\left( \frac{\alpha b+a}{\alpha +1},\frac{\beta d+c}{\beta +1} \right) \notag \\ &\quad\leq \left[ \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }} J_{a+,c+}^{\alpha ,\beta }f(b,d)\right] ^{n}+\left[ \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}J_{a+,d-}^{\alpha ,\beta }f(b,c)\right] ^{n} \notag \\ &\qquad+\left[ \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}J_{b-,c+}^{\alpha ,\beta }f(a,b)\right] ^{n}+\left[ \frac{\Gamma \left( \alpha +1\right) \Gamma \left( \beta +1\right) }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}J_{b-,d-}^{\alpha ,\beta }f(a,c)\right] ^{n} \notag \\ &\quad\leq \frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f^{n}\left( x,y\right) dydx \notag \\ &\qquad+\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f^{n}\left( x,y\right) dydx \notag \\ &\qquad+\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f^{n}\left( x,y\right) dydx \notag \\ &\qquad+\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f^{n}\left( x,y\right) dydx, \end{aligned} \tag{12}\] and so the first and the second inequalities of (11) are proved. For the proof of the last inequality in (11), by changing of the variables \(x=ta+\left( 1-t\right) b\), \(y=sc+\left( 1-s\right) d\) and \(x=\left( 1-t\right) a+tb,\) \(y=\left( 1-s\right) c+sd\) for \(\left( x,y\right) \in \Delta\) and \(\left( t,s\right) \in \left[ 0,1\right] ^{2}\) in (12) respectively\(,\) and by co-ordinate convexity of \(f\), we have \[\begin{aligned} &\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f^{n}\left( x,y\right) dydx \\ &\qquad+\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( b-x\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f^{n}\left( x,y\right) dydx \\ &\qquad+\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( d-y\right) ^{\beta -1}f^{n}\left( x,y\right) dydx \\ &\qquad+\frac{\alpha \beta }{\left( b-a\right) ^{\alpha }\left( d-c\right) ^{\beta }}\int_{a}^{b}\int_{c}^{d}\left( x-a\right) ^{\alpha -1}\left( y-c\right) ^{\beta -1}f^{n}\left( x,y\right) dydx \\ &\quad=\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f^{n}\left( ta+\left( 1-t\right) b,sc+\left( 1-s\right) d\right) dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f^{n}\left( ta+\left( 1-t\right) b,\left( 1-s\right) c+sd\right) dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f^{n}\left( \left( 1-t\right) a+tb,sc+\left( 1-s\right) d\right) dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}f^{n}\left( \left( 1-t\right) a+tb,\left( 1-s\right) c+sd\right) dsdt \\ &\quad\leq \alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}\left[ tsf\left( a,c\right) +t\left( 1-s\right) f\left( a,d\right) +\left( 1-t\right) sf\left( b,c\right) +\left( 1-t\right) \left( 1-s\right) f\left( b,d\right) \right] ^{n}dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}\left[ t\left( 1-s\right) f\left( a,c\right) +tsf\left( a,d\right) +\left( 1-t\right) \left( 1-s\right) f\left( b,c\right) +\left( 1-t\right) sf\left( b,d\right) \right] ^{n}dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}\left[ \left( 1-t\right) sf\left( a,c\right) +\left( 1-t\right) \left( 1-s\right) f\left( a,d\right) +tsf\left( b,c\right) +t\left( 1-s\right) f\left( b,d\right) \right] ^{n}dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}\left[ \left( 1-t\right) \left( 1-s\right) f\left( a,c\right) +\left( 1-t\right) sf\left( a,d\right) +t\left( 1-s\right) f\left( b,c\right) +tsf\left( b,d\right) \right] ^{n}dsdt \\ &\quad=\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}\sum \limits_{k_{1}+k_{2}+k_{3}+k_{4}=n}\frac{n!}{k_{1}!k_{2}!k_{3}!k_{4}!} t^{k_{1}+k_{2}}\left( 1-t\right) ^{k_{3}+k_{4}}s^{k_{1}+k_{3}}\left( 1-s\right) ^{k_{2}+k_{4}} \\ &\qquad\times f^{k_{1}}\left( a,c\right) f^{k_{2}}\left( a,d\right) f^{k_{3}}\left( b,c\right) f^{k_{4}}\left( b,d\right) dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}\sum \limits_{k_{1}+k_{2}+k_{3}+k_{4}=n}\frac{n!}{k_{1}!k_{2}!k_{3}!k_{4}!} t^{k_{1}+k_{2}}\left( 1-t\right) ^{k_{3}+k_{4}}s^{k_{2}+k_{4}}\left( 1-s\right) ^{k_{1}+k_{3}} \\ &\qquad\times f^{k_{1}}\left( a,c\right) f^{k_{2}}\left( a,d\right) f^{k_{3}}\left( b,c\right) f^{k_{4}}\left( b,d\right) dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}\sum \limits_{k_{1}+k_{2}+k_{3}+k_{4}=n}\frac{n!}{k_{1}!k_{2}!k_{3}!k_{4}!} t^{k_{3}+k_{4}}\left( 1-t\right) ^{k_{1}+k_{2}}s^{k_{1}+k_{3}}\left( 1-s\right) ^{k_{2}+k_{4}} \\ &\qquad\times f^{k_{1}}\left( a,c\right) f^{k_{2}}\left( a,d\right) f^{k_{3}}\left( b,c\right) f^{k_{4}}\left( b,d\right) dsdt \\ &\qquad+\alpha \beta \int_{0}^{1}\int_{0}^{1}t^{\alpha -1}s^{\beta -1}\sum \limits_{k_{1}+k_{2}+k_{3}+k_{4}=n}\frac{n!}{k_{1}!k_{2}!k_{3}!k_{4}!} t^{k_{3}+k_{4}}\left( 1-t\right) ^{k_{1}+k_{2}}s^{k_{2}+k_{4}}\left( 1-s\right) ^{k_{1}+k_{3}} \\ &\qquad\times f^{k_{1}}\left( a,c\right) f^{k_{2}}\left( a,d\right) f^{k_{3}}\left( b,c\right) f^{k_{4}}\left( b,d\right) dsdt \\ &\quad=\alpha \beta \sum \limits_{k_{1}+k_{2}+k_{3}+k_{4}=n}\frac{n!}{ k_{1}!k_{2}!k_{3}!k_{4}!}\int_{0}^{1}\int_{0}^{1}\left[ t^{\alpha +k_{1}+k_{2}-1}\left( 1-t\right) ^{k_{3}+k_{4}}+t^{\alpha +k_{3}+k_{4}-1}\left( 1-t\right) ^{k_{1}+k_{2}}\right] \\ &\qquad\times \left[ s^{\beta +k_{1}+k_{3}-1}\left( 1-s\right) ^{k_{2}+k_{4}}+s^{\beta +k_{2}+k_{4}-1}\left( 1-s\right) ^{k_{1}+k_{3}} \right] f^{k_{1}}\left( a,c\right) f^{k_{2}}\left( a,d\right) f^{k_{3}}\left( b,c\right) f^{k_{4}}\left( b,d\right) dsdt \\ &\quad=\alpha \beta \sum \limits_{k_{1}+k_{2}+k_{3}+k_{4}=n}\frac{n!}{ k_{1}!k_{2}!k_{3}!k_{4}!}\left[ \frac{\left( \alpha +k_{1}+k_{2}-1\right) !\left( k_{3}+k_{4}\right) !}{\left( \alpha +n\right) !}+\frac{\left( \alpha +k_{3}+k_{4}-1\right) !\left( k_{1}+k_{2}\right) !}{\left( \alpha +n\right) ! }\right] \\ &\qquad\times \left[ \frac{\left( \beta +k_{1}+k_{3}-1\right) !\left( k_{2}+k_{4}\right) !}{\left( \alpha +n\right) !}+\frac{\left( \beta +k_{2}+k_{4}-1\right) !\left( k_{1}+k_{3}\right) !}{\left( \alpha +n\right) ! }\right] f^{k_{1}}\left( a,c\right) f^{k_{2}}\left( a,d\right) f^{k_{3}}\left( b,c\right) f^{k_{4}}\left( b,d\right) . \end{aligned}\]
Using the multinomial theorem, for an expression of the form \(\left( x_{1}+x_{2}+…+x_{n}\right) ^{n}\) the expansion is given by: \[\left( x_{1}+x_{2}+…+x_{n}\right) ^{n}=\sum \limits_{k_{1}+k_{2}+…+k_{m}=n}\frac{n!}{k_{1}!k_{2}!…k_{m}!} x_{1}^{k_{1}}x_{2}^{k_{2}}…x_{m}^{k_{m}}.\]
Here \(k_{1},\) \(k_{2},\ldots ,k_{m}\) are non-negative integers representing the powers of each term in the expansion, and their sum equals \(n\). Thus, we have the conclusion (11). ◻
Remark 4. Under the assumptions of Theorem 7, if we take \(n=1\), the inequality (11) reduces to the fractional integral inequality for co-ordinated convex functions as in Theorem 5.
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