In this section, using Lemma 1, we established some weighted Ostrowski type inequalities for co-ordinated convex mapping.
First, we define the following mapping
$$\begin{aligned}& \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4};\mathcal{F},w) \\& \quad = \mathcal{F}(\kappa ,\gamma )- \frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,\gamma )\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(\kappa ,v)\,dv\,du \\& \qquad {}+\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,v)\,dv\,du \end{aligned}$$
(2.1)
where
$$ m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})= \int _{\rho _{1}}^{ \rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v)\,dv\,du. $$
Using the change of variables in Lemma 1, we have the following identity:
$$\begin{aligned}& \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4};\mathcal{F},p) \\& \quad = \frac{ ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{0}^{1} \int _{0}^{1} \biggl[ \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr] \\& \qquad {}\times \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}( \varphi ) \bigr)\,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{0}^{1} \int _{0}^{1} \biggl[ \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr] \\& \qquad {}\times \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}( \varphi ) \bigr)\,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{0}^{1} \int _{0}^{1} \biggl[ \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr] \\& \qquad {}\times \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}( \varphi ) \bigr)\,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{0}^{1} \int _{0}^{1} \biggl[ \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr] \\& \qquad {}\times \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}( \varphi ) \bigr)\,d\varphi \,d\psi , \end{aligned}$$
(2.2)
where \(U_{1}(\psi )=\psi \kappa +(1-\psi )\rho _{1}\), \(U_{2}(\psi )=\psi \kappa +(1-\psi )\rho _{2}\), \(V_{1}(\varphi )=\varphi \gamma +(1-\varphi )\rho _{3}\) and \(V_{2}(\varphi )=\varphi \gamma +(1-\varphi )\rho _{4}\).
Theorem 3
Suppose that the mapping w is as in Lemma 1. Moreover, let w is bounded on Δ, i.e., \(\Vert w \Vert _{\infty }:=\sup_{(\kappa , \gamma )\in \Delta } \vert w(\kappa ,\gamma ) \vert \). If \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert \) is a co-ordinated convex function on Δ, then for all \((\kappa ,\gamma )\in \Delta \) we have the following inequality
$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ \Vert w \Vert _{\infty }}{36\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl\{ ( \kappa -\rho _{1} ) ^{2} ( \gamma -\rho _{3} ) ^{2} \biggl[ 4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert \\& \qquad {} +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert \biggr] \\& \qquad {}+ ( \kappa -\rho _{1} ) ^{2} ( \rho _{4}- \gamma ) ^{2} \biggl[ 4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert \\& \qquad {}+2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert \biggr] \\& \qquad {}+ ( \rho _{2}-\kappa ) ^{2} ( \gamma -\rho _{3} ) ^{2} \biggl[ 4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert \\& \qquad {} +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert \biggr] \\& \qquad {}+ ( \rho _{2}-\kappa ) ^{2} ( \rho _{4}- \gamma ) ^{2} \biggl[ 4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert \\& \qquad {} +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert \biggr] \biggr\} , \end{aligned}$$
(2.3)
where the mapping Θ is defined as in (2.1).
Proof
By taking the modulus of the equality (2.2), we have
$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad = \frac{ ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi . \end{aligned}$$
(2.4)
Since \(w(\kappa ,\gamma )\) is bounded on Δ, and \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert \) is co-ordinated convex on Δ, we obtain
$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \quad \leq \Vert w \Vert _{\infty } \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}( \psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \quad \leq ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) \Vert w \Vert _{\infty } \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl[ \psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert +\psi ( 1- \varphi ) \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert \\& \qquad {}+ ( 1-\psi ) \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1}, \gamma ) \biggr\vert + ( 1-\psi ) ( 1- \varphi ) \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert \biggr]\,d\varphi \,d \psi \\& \quad = ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) \Vert w \Vert _{\infty } \\& \qquad {}\times \biggl[ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert + \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert \\& \qquad {} +\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert \biggr] . \end{aligned}$$
(2.5)
Similarly, we have
$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \end{aligned}$$
(2.6)
$$\begin{aligned}& \quad \leq ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) \Vert w \Vert _{\infty } \\& \qquad {}\times \biggl[ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert + \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert \biggr] , \\& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \quad \leq ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) \Vert w \Vert _{\infty } \\& \qquad {}\times \biggl[ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert + \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert \biggr], \end{aligned}$$
(2.7)
and
$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \quad \leq ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) \Vert w \Vert _{\infty } \\& \qquad {}\times \biggl[ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert + \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert \biggr] . \end{aligned}$$
(2.8)
If we substitute the inequalities (2.5)–(2.8) in (2.4), then we obtain the required result (2.3). This completes the proof. □
Corollary 1
Under the same assumption of Theorem 3with \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \vert \leq M\), \(( \kappa ,\gamma ) \in \Delta \), we have the following weighted Ostrowski type inequality
$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{M ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl[ \frac{1}{4}+ \frac{ ( \kappa -\frac{\rho _{1}+\rho _{2}}{2} ) ^{2}}{ ( \rho _{2}-\rho _{1} ) ^{2}} \biggr] \biggl[ \frac{1}{4}+\frac{ ( \gamma -\frac{\rho _{3}+\rho _{4}}{2} ) ^{2}}{ ( \rho _{4}-\rho _{3} ) ^{2}} \biggr] \Vert w \Vert _{\infty }. \end{aligned}$$
Remark 1
If we choose \(w(\kappa ,\gamma )=1\) in Corollary 1, then Corollary 1 reduces to [14, Theorem 3].
Corollary 2
Under the same assumption of Theorem 3with \(\kappa =\frac{\rho _{1}+\rho _{2}}{2}\) and \(\gamma =\frac{\rho _{3}+\rho _{4}}{2}\), we have the following weighted Hermite–Hadamard type inequality
$$\begin{aligned}& \biggl\vert \mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) + \frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,v)\,dv\,du \\& \qquad {}-\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F} \biggl( u, \frac{\rho _{3}+\rho _{4}}{2} \biggr)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v)\mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2},v \biggr)\,dv\,du \biggr\vert \\& \quad \leq \frac{ ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{576} \frac{ \Vert w \Vert _{\infty }}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl\{ 16 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert +4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{3} \biggr) \biggr\vert \\& \qquad {}+4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert +4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{4} \biggr) \biggr\vert \\& \qquad {}+4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert \biggr\} \\& \quad \leq \frac{ ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{64} \frac{ \Vert w \Vert _{\infty }}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert \biggr] . \end{aligned}$$
Theorem 4
Let w be as in Theorem 3. If \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert ^{q}\) is a co-ordinated convex function on Δ, then for all \((\kappa ,\gamma )\in \Delta \), we have the following inequality
$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ \Vert w \Vert _{\infty }}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})(p+1)^{\frac{2}{p}}} \\& \qquad {}\times \biggl\{ ( \kappa -\rho _{1} ) ^{2} ( \gamma -\rho _{3} ) ^{2} \\& \qquad {}\times \biggl( \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr] \biggr) ^{\frac{1}{q}} \\& \qquad {}+ ( \kappa -\rho _{1} ) ^{2} ( \rho _{4}- \gamma ) ^{2} \\& \qquad {}\times \biggl( \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr] \biggr) ^{\frac{1}{q}} \\& \qquad {}+ ( \rho _{2}-\kappa ) ^{2} ( \gamma -\rho _{3} ) ^{2} \\& \qquad {}\times \biggl( \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr] \biggr) ^{\frac{1}{q}} \\& \qquad {}+ ( \rho _{2}-\kappa ) ^{2} ( \rho _{4}-\gamma ) ^{2} \\& \qquad {} \times \biggl( \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr] \biggr) ^{ \frac{1}{q}} \biggr\} , \end{aligned}$$
(2.9)
where the mapping Θ is defined as in (2.1), and \(\frac{1}{p}+\frac{1}{q}=1\).
Proof
Using the well-known Hölder inequality in (2.4), we obtain
$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}( \psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \biggr) ^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}( \psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}( \psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \biggr) ^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}( \psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}( \psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \biggr) ^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}( \psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}( \psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \biggr) ^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}( \psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}}. \end{aligned}$$
(2.10)
Since w is bounded on Δ, we have
$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \\& \quad \leq \Vert w \Vert _{ \infty }^{p} \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}( \varphi )}\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \\& \quad = \Vert w \Vert _{\infty }^{p} ( \kappa -\rho _{1} ) ^{p} ( \gamma -\rho _{3} ) ^{p} \int _{0}^{1} \int _{0}^{1}\varphi ^{p}\psi ^{p}\,d\varphi \,d\psi \\& \quad = \frac{ ( \kappa -\rho _{1} ) ^{p} ( \gamma -\rho _{3} ) ^{p}}{(p+1)^{2}} \Vert w \Vert _{\infty }^{p}. \end{aligned}$$
(2.11)
Similarly, we get
$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \leq \frac{ ( \kappa -\rho _{1} ) ^{p} ( \rho _{4}-\gamma ) ^{p}}{(p+1)^{2}} \Vert w \Vert _{\infty }^{p}, \end{aligned}$$
(2.12)
$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \leq \frac{ ( \rho _{2}-\kappa ) ^{p} ( \gamma -\rho _{3} ) ^{p}}{(p+1)^{2}} \Vert w \Vert _{\infty }^{p}, \end{aligned}$$
(2.13)
and
$$ \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \leq \frac{ ( \rho _{2}-\kappa ) ^{p} ( \rho _{4}-\gamma ) ^{p}}{(p+1)^{2}} \Vert w \Vert _{\infty }^{p}. $$
(2.14)
On the other hand, as \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert ^{q}\) is a co-ordinated convex function on Δ, we obtain
$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}( \varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \end{aligned}$$
(2.15)
$$\begin{aligned}& \quad \leq \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr] , \\& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}( \varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \end{aligned}$$
(2.16)
$$\begin{aligned}& \quad \leq \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr] , \\& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}( \varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \\& \quad \leq \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr] , \end{aligned}$$
(2.17)
and
$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}( \varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \\& \quad \leq \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr] . \end{aligned}$$
(2.18)
If we substitute the inequalities (2.11)–(2.18) in (2.10), then we obtain the required inequality (2.9). □
Corollary 3
Under the same assumption of Theorem 4with \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \vert \leq M\), \(( \kappa ,\gamma ) \in \Delta \), then we have the following weighted Ostrowski type inequality
$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{4M ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})(p+1)^{\frac{2}{p}}} \biggl[ \frac{1}{4}+ \frac{ ( \kappa -\frac{\rho _{1}+\rho _{2}}{2} ) ^{2}}{ ( \rho _{2}-\rho _{1} ) ^{2}} \biggr] \biggl[ \frac{1}{4}+ \frac{ ( \gamma -\frac{\rho _{3}+\rho _{4}}{2} ) ^{2}}{ ( \rho _{4}-\rho _{3} ) ^{2}} \biggr] \Vert w \Vert _{\infty }. \end{aligned}$$
Remark 2
If we choose \(w(\kappa ,\gamma )=1\) in Corollary 3, then Corollary 3 reduces to [14, Theorem 4].
Corollary 4
Under the same assumption of Theorem 4with \(\kappa =\frac{\rho _{1}+\rho _{2}}{2}\) and \(\gamma =\frac{\rho _{3}+\rho _{4}}{2}\), then we have the following weighted Hermite–Hadamard type inequality
$$\begin{aligned}& \biggl\vert \mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) + \frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,v)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F} \biggl( u, \frac{\rho _{3}+\rho _{4}}{2} \biggr)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v)\mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2},v \biggr)\,dv\,du \biggr\vert \\& \quad \leq \frac{ \Vert w \Vert _{\infty } ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{2^{4+\frac{2}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})(p+1)^{\frac{2}{p}}} \\& \qquad {}\times \biggl\{ \biggl( \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{3} \biggr) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \biggl( \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{4} \biggr) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \biggl( \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{3} \biggr) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \biggl( \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{4} \biggr) \biggr\vert ^{q} \\& \qquad {} + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \biggr\} . \end{aligned}$$
Theorem 5
Let w be as in Theorem 3. If the function \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert ^{q}\) is a co-ordinated convex function on Δ, then for all \((\kappa ,\gamma )\in \Delta \) and \(q\geq 1 \), we have the following inequality
$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ ( \kappa -\rho _{1} ) ^{2} ( \gamma -\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{4\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ^{2} ( \rho _{4}-\gamma ) ^{2} \Vert w \Vert _{\infty }}{4\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ^{2} ( \gamma -\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{4\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ^{2} ( \rho _{4}-\gamma ) ^{2} \Vert w \Vert _{\infty }}{4\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}, \end{aligned}$$
(2.19)
where the mapping Θ is defined as in (2.1).
Proof
By utilizing the power mean inequality in (2.4), we obtain
$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}( \psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \,d\varphi \,d \psi \biggr) ^{1-\frac{1}{q}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}( \varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}( \psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \,d\varphi \,d\psi \biggr) ^{1-\frac{1}{q}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}( \varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}( \psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \,d\varphi \,d \psi \biggr) ^{1-\frac{1}{q}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}( \varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}( \psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \,d\varphi \,d\psi \biggr) ^{1-\frac{1}{q}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}( \varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \quad \leq \frac{ ( \kappa -\rho _{1} ) ^{2} ( \gamma -\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{4^{1-\frac{1}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ^{2} ( \rho _{4}-\gamma ) ^{2} \Vert w \Vert _{\infty }}{4^{1-\frac{1}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ^{2} ( \gamma -\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{4^{1-\frac{1}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ^{2} ( \rho _{4}-\gamma ) ^{2} \Vert w \Vert _{\infty }}{4^{1-\frac{1}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}}. \end{aligned}$$
(2.20)
Since \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert ^{q}\) is a co-ordinated convex function on Δ, we have the following inequalities
$$\begin{aligned}& \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \end{aligned}$$
(2.21)
$$\begin{aligned}& \quad \leq \biggl( \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}, \\& \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \end{aligned}$$
(2.22)
$$\begin{aligned}& \quad \leq \biggl( \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}, \\& \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \quad \leq \biggl( \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}, \end{aligned}$$
(2.23)
and
$$\begin{aligned}& \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \quad \leq \biggl( \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}. \end{aligned}$$
(2.24)
By utilizing the equalities (2.21)–(2.24) in (2.20), we obtain the desired inequality (2.19). This completes the proof. □
Remark 3
Under the same assumption of Theorem 5 with \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \vert \leq M\), \(( \kappa ,\gamma ) \in \Delta \), then Theorem 5 reduces to Corollary 1.
Corollary 5
Under the same assumption of Theorem 5with \(\kappa =\frac{\rho _{1}+\rho _{2}}{2}\) and \(\gamma =\frac{\rho _{3}+\rho _{4}}{2}\), we have the following weighted Hermite–Hadamard type inequality
$$\begin{aligned}& \biggl\vert \mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) + \frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,v)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F} \biggl( u, \frac{\rho _{3}+\rho _{4}}{2} \biggr)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})}\int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v)\mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2},v \biggr)\,dv\,du \biggr\vert \\& \quad \leq \frac{ ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{64\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl[ \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{3} \biggr) \biggr\vert ^{q} \\& \qquad {}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \rho _{4} \biggr) \biggr\vert ^{q} \\& \qquad {}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \rho _{3} \biggr) \biggr\vert ^{q} \\& \qquad {} + \biggl(\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr) \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \rho _{4} \biggr) \biggr\vert ^{q} \\& \qquad {} +\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$