Some Hardy-type integral inequalities involving functions of two independent variables

Abstract

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.