# lower bound of the objective function of a constrained optimization problem

Consider the following constrained optimization problem \begin{align*} &\underset{p,\theta}{\max}~R(p,\theta)\triangleq\frac{qp\theta}{1-q}+p(1-\theta)(1-p)\\ &\mbox{s.t. }~q = 1-\frac{c+\theta p}{\theta +q(1-\theta)}\\ &~~~~p\geq \frac{c}{q(1-\theta)}\\ &~~~~\theta\in[0,1]\\ &~~~~q\in[0,1]\\ &~~~~p\geq 0 \end{align*} Let $$R^*(p^*,\theta^*)$$ be the optimal objective function value of this optimization problem.

How to show that $$R^*(p^*,\theta^*)>(1-\sqrt{c})^2$$ for $$c\in[0,1/4)$$?

I have done numerical analysis, and find that $$R^*$$ is decreasing in $$c$$ and when $$c\rightarrow~0$$, $$R^*=1.25$$ and when $$c=0.25$$, then $$R^*=0.25$$. However, due to the fact that $$q$$ does not have an explicit expression (I can solve the equation of $$q$$ and get two solutions, but that does not seem to help), I couldn't come up with a way to prove this claim. I can prove the case of $$c\rightarrow~0$$, but for general $$c\in[0,1/4)$$, it seems to be very challenging.