What asymptotic lower bounds are known on the number of primes between $n$ and $2n$, call it $\Pi(n) = \pi(2n)-\pi(n)$? Obviously Bertrand's Theorem is the statement that $\Pi(n) \geq 1$ for all $n$, and the PNT says that $\Pi(n) \approx \frac{n}{\log n}$ but doesn't provide a lower bound.
It seems like Hoheisel's result on prime gaps that there exists a $\theta \lt 1$ such that $\pi(x+x^\theta)-\pi(x) \approx \frac{x^\theta}{\log x}$ (as seen at http://en.wikipedia.org/wiki/Guido_Hoheisel ) should give a lower bound of the form $\Pi(n) \geq k\frac{n}{\log n}$ for some $k$ by stacking up $O(x^{1-\theta})$ such intervals; more precisely, if we take $c=2^{\theta}$ and $f(x) = x+x^\theta$, then since $x^\theta \leq cn^\theta$ for $x\leq 2n$, we have $f^{\circ t}(n) = f^{\circ(t-1)}(n) + \left(f^{\circ(t-1)}(n)\right)^\theta\leq f^{\circ(t-1)}(n) +cn^\theta\leq n+ctn^\theta$ as long as $ctn^\theta \lt n$ - or in other words, as long as $t\lt t_0 = c^{-1}n^{1-\theta}$, and therefore $\begin{align*} \Pi(n) &= \pi(2n)-\pi(n) \\ &\geq \sum_{i=0}^{t_0-1} \left(\pi(f^{\circ (i+1)}(n))-\pi(f^{\circ i}(n))\right) \\ &\geq \sum_{i=0}^{t_0-1}\frac{(f^{\circ i}(n))^\theta}{\log f^{\circ i}(n)} \\ &\geq \sum_{i=0}^{t_0-1}\frac{n^\theta}{\log n} \\ &= t_0 \frac{n^\theta}{\log n} \\ &= c^{-1}\frac{n}{\log n} \end{align*}$
Is this reasoning correct? It makes sense to me, but I'm surprised I've never seen such an explicit lower bound given before. While I'm more interested in the asymptotics, I'm also curious about the precise constants; if this is a known lower bound, is anything known about explicit $k$ such that $\Pi(n) \geq k\frac{n}{\log n}$ for all $n$ (or even all $n\gt n_0$)?