@MateuszKwaśnicki I may not have made it clear enough but what I mean is that $Y$ is some given random variable that you don't get to choose. (Of course condition (ii) implies $\mathbb{E}[Y^p] < \infty$ so you can always rewrite a constant $C$ as the product of $\mathbb{E}[Y^p]$ and a new constant $C'$.

I am feeling so embarrassed that I have missed this observation...thank you so much for your patience and again your counter-example.

Oh I have asked a dumb question. I actually want to ask if a lower bound $f(t) \ge \frac{1}{Ct}$ is possible.

Thanks for the nice counter-example. Would it still be possible to establish the upper bound $f(t) \le \frac{C}{t}$ though?

@MateuszKwaśnicki Thanks, I am looking forward to your answer.

What about bounds? Is it possible to show that there exists some $C > 0$ such that $\frac{1}{Ct} \le f(t) \le \frac{C}{t}$ for $t$ sufficiently large?

@Venkataramana I could be wrong, but aren't you suggesting the converse of L'Hospital's rule (if the ratio of two functions has a limit, then the ratio of the derivatives has the same limit), which does not seem to be true in general? I have no control over $o(\log t)$ and just can't say much about its derivative, and I am not sure how L'Hospital may be applied.

In fact before posting the question I was thinking about the same thing. The problem (which might be stupid, since I am not very strong in complex analysis) I had in mind was: how do I know $f$ is holomorphic on $\{z \in \mathbb{C}: 0 \le Re ~z < 1/2\}$?