I'm curious to know if it is in the literature a similar/analogous statement about Ramanujan primes (this Wikipedia Ramanujan prime or [1]), than the known as Bonse's inequality for prime numbers (see from Wikipedia Bonse's inequality.) Any case, if isn't in the literature I propose to get a remarkable improvement of my humble proposal (next claim).
For integers $k\geq 1$ let $R_k$ the $k$-th Ramanujan prime (sequence A104272 from the OEIS).
Claim. For each integer $n\geq 4$ the inequality $$R_{n+1}^2<\prod_{k=1}^n R_k\tag{1}$$ holds.
Sketch of proof. We use Theorem 2 (that is the first theorem of section 3) from [1]. Then it is sufficies to prove $(4(n+1)^2)^2<4^n n!$ for enough large $n$, and maybe add few computational examples as companion of this theoretical reasoning. $\square$
Question. I would like to know if it is possible/feasible to get a remarkable improvement of previous inequality $(1)$ for a positive arithmetic function $f:\mathbb{N}\to\mathbb{R}_{>0}$, satisfying (that is increasing) $f(n+1)\geq f(n)$ and also $f(n)\geq 2$ let's say for all $n\geq n_0\geq 1$ (here, thus $n_0$ is a choice of a suitable constant in your inequality) with the purpose that $$(R_{n+1})^{f(n)}<\prod_{k=1}^n R_k,\tag{2}$$ holds for all $n\geq n_0$. Many thanks.
Then explain your reasoning, or hints to get a remarkable improvement of $(1)$ of the form $(2)$ satisfying $\forall n\geq n_0$ for a suitable choice of $n_0$ and your arithmetic function $f(n)$.
I'm trying to get an answer, $n_0>1$, although I am not clear how to get a statement $(2)$ with a good mathematical content from the proposal
$$R_{n+1}^{W(n)}<\prod_{k=1}^n R_k,$$ where $W(x)$ is the principal/main branch of the Lambert $W$ function. I think that should be a nice proposal for some function $f(x)$ of slow increasing, with the goal to get an improvement of the proposal in previous Claim as was it asked in my question.
References:
[1] Jonathan Sondow, Ramanujan Primes and Bertrand's Postulate, The American Mathematical Monthly, Vol. 116, No. 7 (2009), pp. 630-635.