Won't turn on, no vibration when in silent, but detected by iTunes

Has your screen been replaced recently or has any other repair taken place on your device? Has it been dropped or otherwise damaged?

If you can't see anything on the screen but iTunes recognizes your phone, then you have a backlight issue. Shine a bright flashlight on the screen after pushing the Home button; do you see a very dim image? If you can, then this could be a screen problem or a backlight driver circuit issue on the logic board.

The only way to isolate that is to try a known-good screen. If the problem is on the logic board, then this will require a micro-soldering repair and is not DIY.