An easy supremum problem

Question

Let $f$ be a positive decreasing function. What can we say about the following supremum. $$ \sup_{t>1}\ln t\,f(t). $$ More generally what can we say about the supremum of product of an increasing function and a decreasing function?

Answer 1

Inconclusive.

If $f(t) := 1/(\log\log t)$ for all suitable $t$, then $\log(t) f(t) \to \infty$; wheraes if $f(t) := 1/t$ for all suitable $t$, then $\log(t) f(t) \to 0$. In both cases $f > 0$ eventually and is decreasing.