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given $\alpha_i > 0 $, $\beta_i \geq 1$ with $i=1$ or $2$

$R_i(t)=e^{-\alpha_i t^{\beta_i}}$ is Weibull survival function

$\phi_i(t)=\int_{t}^{\infty}R_i(u)du$

$f(x,y)=R_1(x)R_1(y)\phi_2(x+y)$

$g(x,y)=R_2(x)R_2(y)\phi_1(x+y)$

I would like to know if the following inequality involving symmetric bilinear forms holds:

$f \cdot \frac{\partial^2 g}{\partial x \partial y}+g \cdot \frac{\partial^2 f}{\partial x \partial y}- \frac{\partial f}{\partial x} \cdot \frac{\partial g}{\partial y} - \frac{\partial f}{\partial y} \cdot \frac{\partial g}{\partial x} \leq 0$

Reference: the last inequality appeared in (Aaradji 2015) where the author dealt with total positive functions of order 2. In my question, $f(t)$ is of $RR_2$ (reverse rule of order 2, see Karlin 1980) so does $g(t)$. I would like to know if the sum of $f$ and $g$ is still $RR_2$.

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