Ramanujan primes in short intervals

Question

I'm curious to know if it is in the literature a similar/analogous statement about Ramanujan primes (this Wikipedia Ramanujan prime or [1]) in short intervals than those that refers the Wikipedia Bertrand's postulate for prime numbers due to Pierre Dusart or Baker, Harman and Pintz:

There is at least a Ramanujan prime $R$ in the interval $$x<R\leq x+x\cdot f(x)\tag{1}$$ or, say, in the interval $$x−x\cdot f(x)\leq R\leq x,\tag{2}$$ for a suitable function $f(x)$, and for all $x\geq x_0$ being $x_0$ a suitable constant (it is your choice the value of this constant).

Question. If you know it from the literature please refer, answering as a reference request, the literature and I try to search and read those statements from the literature. In case that isn't in the literature, can you provide a statement for Ramanujan primes in short intervals of the form $(1)$ or $(2)$? Many thanks.

I think that this is an interesting question, please feel free to add your feedback in comments.

References:

[1] Jonathan Sondow, Ramanujan Primes and Bertrand's Postulate, The American Mathematical Monthly, Vol. 116, No. 7 (2009), pp. 630-635.

Answer 1

According to Wikipedia, for $n>1$ we have $p_{2n}<R_n<p_{3n}$, where $p_n$ is the $n$-th prime number. But $p_{2n}>2n(\log (2n)+\log\log (2n)-1)$ and $p_{3n}<3n(\log (3n)+\log\log (3n))$ for $n\ge 3$, see Wikipedia. Thus if $x\le 2n(\log (2n)+\log\log (2n)-1)$ then between $x$ and $3n(\log (3n)+\log\log (3n))$ there is a Ramanujan prime. This should provide a bound $f(x)\approx\tfrac 12$.