A semiprime $s$ is a positive integer that is the product of two prime numbers, see Semiprine the encyclopedia Wikipedia. A well-known inequality, with applications, that involves prime numbers is the named Bonse's inequality, I add as reference from the Wikipedia Bonse's inequality.
I wondered if it is possible to create a similar inequality where $s_k$ denotes the $k$-th semiprime (thus the sequence A001358 from the OEIS)
$$m(n)\cdot (s_{n+1})^{a}<\left(\prod_{k=1}^n s_k\right)^b$$ for a suitable arithmetic function (or sequence) $m(n)$ and constants $a$ and $b$ (I've added this function $m(n)$ and constant $b$ with the purpose to provide flexibility in the research of the inequality that we evoke).
Question. How to get a sharper inequality* involving semiprimes $s_k$ $$(s_{n+1})^{a}<\frac{1}{m(n)}\left(\prod_{k=1}^n s_k\right)^b\tag{1}$$ that holds $\forall n>N$ for a suitable choice of $N$, and for a suitable choice of a function $m(n)>0$ and constants $a$ and $b$? Many thanks
*If in your investigations you get a remarkable inequality, or asymptotic inequality, that does not fit exactly to the previous type of inequality $(1)$ I think that it is reasonable that you can to feel free to add it as an answer because I am asking for a good version of a Bonse's inequality for semiprimes.
With the words a sharper inequality I mean that your inequality of the type $(1)$ have good mathematical content/meaning, that it is a good Bonse's inequality for semiprimes. To emphasize we take yours $m(n)$ as a positive arithmetic function, thus $m(n)>0$ for all $n>N$.
