\begin{equation} \begin{split} S(t) = \frac{1}{\int_{0}^{1}\prod_{m=1}^M\left( \prod_{i=1}^{n_m} (t_{mi}+yx)^{\beta-1}\right)y^{c-1}(1-y)^{d-1}\frac{ \Gamma(\sum_{m=1}^{M}n_m+a)}{\left[\sum_{m=1}^{M}\sum_{i=1}^{n_m}\left( (t_{mi}+yx)^{\beta}-(yx)^{\beta} \right)+b \right]^{\sum_{m=1}^{M}n_m+a}}dy} \\ \times \int_{0}^{1} \left( \prod_{m=1}^M \prod_{i=1}^{n_m} (t_{mi}+yx)^{\beta-1} \right) y^{c-1} (1-y)^{d-1} \\ \times \frac{\Gamma\left(\sum_{m=1}^{M} n_m + a\right)}{\left[ \sum_{m=1}^{M} \sum_{i=1}^{n_m} \left( (t_{mi}+yx)^{\beta} - (yx)^{\beta} \right) + b + \left( (t_{i}+yx_j)^{\beta} - (yx_j)^{\beta} \right) \right]^{\sum_{m=1}^{M} n_m + a}} \, dy \end{split} \end{equation}
I use the Bayesian method to get a survival equation, which is complex and challenging. I want to show this function's upper and lower in $a,\ b,\ c>0$ where $\beta>0$ is constant. I have been trying to take the derivative to see if the function either decreases or increases, but I can't get the derivative. Is there any idea or help to prove the lower and upper bound?
