Is there an upper bound for $\pi (n)-\pi (n/2)$? Is there a nice upper bound for $\pi (n)-\pi (n/2)$ where $\pi$ is the prime counting function?


$\frac{n}{ln(n)}<\pi(n)<1.25506\frac{n}{ln(n)}$ for $n\geq 17$
See Rosser, Schoenfeld 1961 corollary 1. 
Provides:
$\pi(n)-\pi(\frac{n}{2})<\left(1.25506\frac{n}{ln(n)}-\frac{\frac{n}{2}}{ln(n/2)} \right)$

