To make sure that this is not software related, try these steps:
1. Force restart the phone by holding the sleep and home button for about 10sec, until the Apple logo comes back again. You will not lose data.
2. In case this does not work, set it back to factory settings, without using the latest backup. Test the screen. If it works, restore from the backup.
3. If factory resetting does not help, get the phone checked by Apple:
Start a service request
Find an Apple Authorized Service Provider
Make an Apple Store appointment