Let $x_1,x_2,x_3,\ldots,x_S$ be numbers with $x_i>-1$ for all $i$ and $x_k<0$ for some $k$.
How can one show that \begin{equation} \inf_{s\in[1,S]}\inf_{t\in[1,s]}\prod_{i=t}^s (1+\frac{1}{2}x_i) < \inf_{s\in[1,S]}\inf_{t\in[1,s]}\prod_{i=t}^s (1+\frac{1}{4}x_i) \end{equation} This seems to instinctively be obvious, because the "most destructive path" surely must be a bit less destructive when we reduce the "destruction" from 1/2 to 1/4, but I'm not sure how to formalize this thought.
Update: Try also to generalize this for not just 1/2 and 1/4, but for any number $q$ and any other number $p ., with $0