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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...

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  • $\begingroup$ It seems something is wrong, because if $\alpha, x>0$ and $\beta>1$ then an integral $\int_{x}^\infty \alpha t^\beta dt$ diverges. $\endgroup$ – Alex Ravsky Nov 27 '18 at 18:42
  • $\begingroup$ Thank you for your comments Alex Ravsky. In fact I forgot the exponential... $\endgroup$ – Xingheng Liu Dec 6 '18 at 19:27

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