Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following:

Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\cdots a_n)^{1/n}+\beta a_1?$$

This article derives inequalities of above sort, with $\alpha=1$ and $\beta =1/n$, but with $\sum_{1\le i<j\le n}|a_i-a_j|$ instead of $a_1$, multiplied with $\beta$. Also their method does not seem to be applicable to address my question. Is there any known result on this or is this an open question? Any ideas regarding how should one proceed? Thanks in advance.