Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$? Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$?
Where $\pi(n)$ is the prime counting function. Using PMT it seems asymptotically $\ p_n^{\pi(n)} \leq x^n$ where $e \leq x$

Yes, 
Asymptotically you have
$$(p_n)^{\pi(n)} \leq n^{n/\log n} = e^n$$
