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.