As a matter of fact, P. L. Chebyshev knew already that for any $\epsilon > \frac{1}{5}$, there exists an $n(\epsilon) \in \mathbb{N}$ such that for all $n\geq n(\epsilon),$
$\pi((1+\epsilon)n)-\pi(n)>0.$
In [2], one can find a short report on the problem of determining explicit values for the smallest $n(\epsilon)$ explicitly once that $\epsilon$ has been fixed.
References
[1] P. L. Chebyshev. Mémoire sur les nombres premiers. Mémoires de l'Acad. Imp. Sci. de St. Pétersbourg, VII, 1850.
[2] H. Harborth & A. Kemnitz. Calculations for Bertrand's Postulate. Mathematics Magazine, 54 (1), pp. 33-34.
