In this paper, we give some new generalizations to the Hardy-type integral inequalities for functions of two variables by using weighted mean operators \(S_{1}:=S_{1}^{w}f\) and \(S_{2}:=S_{2}^{w}f\) defined by

$$\begin{aligned}S_{1}(x,y)=\displaystyle \frac{1}{W(x)W(y)}\int _{\frac{x}{2}}^{x}\int _{\frac{y }{2}}^{y}w(t)w(s)f(t,s)dsdt,\end{aligned}$$

and

$$\begin{aligned}S_{2}(x,y)=\displaystyle \int _{\frac{x}{2}}^{x}\int _{\frac{y}{2}}^{y}\frac{ w(t)w(s)}{W(t)W(s)}f(t,s)dsdt,\end{aligned}$$

with

$$\begin{aligned}W(z)=\displaystyle \int _{0}^{z}w(r)dr\quad for\, \,z\in (0,+\infty ),\end{aligned}$$

where w is a weight function.

中文翻译：

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涉及两个自变量函数的一些Hardy型积分不等式

在本文中，我们通过使用加权平均算子\（S_ {1}：= S_ {1} ^ {w} f \）和\（S_ {来为两个变量的函数提供Hardy型积分不等式。2}：= S_ {2} ^ {w} f \）由

$$ \ begin {aligned} S_ {1}（x，y）= \ displaystyle \ frac {1} {W（x）W（y）} \ int _ {\ frac {x} {2}} ^ {x } \ int _ {\ frac {y} {2}} ^ {y} w（t）w（s）f（t，s）dsdt，\ end {aligned} $$

和

$$ \ begin {aligned} S_ {2}（x，y）= \ displaystyle \ int _ {\ frac {x} {2}} ^ {x} \ int _ {\ frac {y} {2}} ^ {y} \ frac {w（t）w（s）} {W（t）W（s）} f（t，s）dsdt，\ end {aligned} $$

与

$$ \ begin {aligned} W（z）= \ displaystyle \ int _ {0} ^ {z} w（r）dr \ quad for \，\，z \ in（0，+ \ infty），\ end {已对齐} $$

其中w是权重函数。