Screen won't turn on

Since you are able to connect and sync with your iTunes account and, thus, have presumably backed up your data, I would try to "Restore" your phone from within iTunes. This will restore it to factory settings, at which point you can re-sync the phone. If a Restore doesn't do the trick, then schedule an appointment at the Genius Bar at the nearest Apple Retail Store (hopefully there is one near you). If it is still under warranty you could also call AppleCare.