Let, $f(x),g(x),f_1(x),g_1(x)$ are positive real valued bounded and continuous functions on domain of non-negative reals and also having range between $0$ and $1$. And, also, $f_1(x),g_1(x)$ are decreasing function.
Now, if $\sup(f(x))\ge\sup(g(x))$ with $f_1(x)\ge g_1(x)$ then is it true that $\sup(f(x)f_1(x))\ge\sup(g(x)g_1(x))$?
If, $f_1(x),g_1(x)$ are constant functions, then above implication is true.
Is this true for non constant functions $f_1(x),g_1(x),f(x),g(x)$?
There is the following counterexample. For each $x\ge 0$ put $f(x)=\frac{2-e^{-x}}{3}$, $g(x)=\frac 12$, and $f_1(x)=g_1(x)=\frac{1+e^{-x}}{3}$. Then $$\sup f(x)=\frac 23>\frac 12=\sup g(x),$$ but $$\sup f(x)f_1(x)=\frac 14<\frac 13=\sup g(x)g_1(x).$$