Sum of product of two sequences

Question

Let $\{a_n\}$ be such that $\tfrac{1}{m}\sum_{n=1}^m a_n$ converges absolutely to some finite constant $a_0$. The $\lim\limits_{m\to\infty}\tfrac{1}{m}\sum_{n=1}^m \tfrac{a_n}{n}$ is trivially upper bounded by $a_0$. Does there exist a tighter upper bound?

Answer 1

No. Consider the sequence with $a_1=1$ and $a_i=0$ for $i>1$. Then we have $$\lim_{m\to\infty}\frac{1}{m}\sum_{n=1}^ma_n=0$$ and $$\lim_{m\to\infty}\frac{1}{m}\sum_{n=1}^m\frac{a_n}{n}=0$$ as well so there is no tighter upper bound.