This question is a cross-post from MSE, cause I didn't get any answer there. I hope it is well suited for MO:
Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and $\displaystyle{\overline{x}_{n}:=\dfrac{1}{n}\sum_{i=1}^{n}x_{i}}$. Suppose furthermore that $\forall\varepsilon\gt 0, \ \ n.\overline{x}_{n}\ll_{\varepsilon}n^{1+\varepsilon}$. Does it entail that $x_{n}\ll \overline{x}_{n}^{2}$?
Thanks in advance.
Edit: I think I must clear up my idea here. Writing $m_{n}:=\dfrac{x_{1}+x_{n}}{2}$ and using the inequality of arithmetic and geometric means, one gets $x_{1}.x_{n}\le m_{n}^{2}$. My idea is that, under the growth condition above, $m_{n}$ is roughly of the same order of magnitude as $\overline{x}_{n}$.