exponential upper bound on sum of exponentials

Question

For what value of $c_3$ can I guarantee that $(a+b)\exp(-c_3\theta)>a\exp(-c_1\theta)+b\exp(-c_2\theta)$ where $a,b,c_i>0$

Answer 1

I assume you want $\theta\gt0$? For an unbounded $\theta \in \mathbb R$, no solution exists. For $\theta\gt0$, the following conditions are both sufficient and necessary: $$c_3 \leq c_1\\c_3\leq c_2\\c_3\lt c_1+c_2$$ It is obvious that the statement is satisfied in this case. On the other hand, if $c_1<c_3$ then there exists a $\theta>0$ such that $(a+b)\exp(-c_3\theta)<a\exp(-c_1\theta)<a\exp(-c_1\theta)+b\exp(-c_2\theta)$