Try to force restart the phone by holding the sleep and home button for about 10sec, until the Apple logo comes back again. After that the phone should start up normally. If it does not, try to connect in recovery mode to your computer, explained in the blue linked article.
If still no luck, get it checked by Apple or an Apple Authorized Service Provider next to your place, the problem might be hardware related.
If your iPhone doesn't respond or won't turn on - Apple Support