Minimization of Convex Function Derived from Moment Generating Functions

Question

Consider the following optimization problem: $$\min_{t > 0} \frac{\prod_{i = 1}^n 1-p+pe^{c_it}}{e^{at}}$$ where $a, c_i \in \mathbb{R}$ and $p \in (0,1).$ How do I find a neat expression for the minimum of the function? My attempt so far was to take the logarithm (since the location of the minimum is the same), compute it's derivative and set it to zero in order to find the argument of the minimum. However, this gave me $$\sum_{i=1}^n \left(\frac{pc_ie^{c_it}}{1-p+pe^{c_it}}\right) = a$$ which I'm unable to solve for $t.$

Answer 1

The correct sum would actually be $$\sum_{i=1}^n \frac{pc_ie^{c_it}}{1-p+pe^{c_it}}=an$$ since summing over a constant $a$ is just multiplying it by $n$.

From then on, the objective would be to find a $t$ such that the terms being summed over are a constant. I'm not quite sure on how you would proceed from here without further information on $c_i$.