define R(x) as the weibull survival function
$R_1(t)=e^{-\alpha_1 t^{\beta_1}}$
$R_2(t)=e^{-\alpha_2 t^{\beta_2}}$
with $\alpha_1, \alpha_2 > 0 $ and $\beta_1, \beta_2 >1 $
$\phi_1(t)=\int_{t}^{\infty}R_1(t) \mathrm{d}t$
$\phi_2(t)=\int_{t}^{\infty}R_2(t) \mathrm{d}t$
I am trying to prove the following inequality (verified by matlab):
$R_1(x)R_1(y)\phi_2(x+y)\phi_1(0)+R_2(x)R_2(y)\phi_1(x+y)\phi_2(0) \leq R_1(x)R_2(y)\phi_2(x)\phi_1(y)+R_2(x)R_1(y)\phi_1(x)\phi_2(y)$
where $x,y \geq 0$
p.s. i found that for the function $\phi$, the following inequality holds: $\phi(x)\phi(y)\geq \phi(x+y)\phi(0)$ but i dont have other ideas...