My iPhone 4s keeps turning itself on.

check out a couple of different things. On clear our multi tasking you can do this by being on the home screen and double tapping the home button twice. a line of application will show up at the bottom. Hold down on one of them and when you see the minus sign tap that on all of them. If it is still turning off randomly it may be software related and at that point i would suggest making a back up in itunes and then restoring make to factory setting

no but that is turning OFF, what about turning on!? Mine does this too!