# sign of a Symmetric bilinear form

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 $$RR_2$$.