If the phone is fully charged and still does not start, try a complete re-establishing. Press the "Home" and "Power" buttons simultaneously and hold them down. The restoration process should begin. You will know that the process has begun when the Apple symbol on the display.
If the iPhone still does not turn on, try to connect the phone to your computer using the USB cable from the iPhone. iTunes should recognize the device and turn it on.
last option is to contact Apple.com / support / iphone. Enter the serial number of your phone and a description of the problem. Support for Apple will send you a number to contact if the problem is not solved online.