My iphone 4s will not stay powered off

When I power off my Iphone, the screen goes black for a few seconds, and then the apple logo appears and the phone powers back on. I have tried resetting it (I think) by holding both the lock button and the home screen button at the same time, but the problem continues. The phone indicates that it is up to date on all software.

The phone also seems to be draining battery charge at a faster rate. I purchased the phone in December 2011, so it is not that old.

Any suggestions.