First thing I would recommend is to find out if the device is still under warranty. It sounds like it might need to be replaced. If the battery is low instead of an Apple Logo, it will come up with a red battery and a Charge Indicator.So the fact that it's coming up with an Apple shows that the battery has some charge, and that it is trying to startup normally.
If it fails and turns off, this seems to indicate that there is something wrong with the way that the phone is turning on. About the only thing that you can do is to plug it into iTunes, and hope that you can perform a restore.In order to attempt a restore, and because the device won't turn on, I believe one of your only options would be to put the device into DFU mode. (See Following Video) youtube.com/v/-KmHNSE3CZI This should bypass the device trying to startup, and put it into a recovery mode that iTunes can handle.
The intent / goal would be to get the computer to recognize it, and allow iTunes to perform a Restore. A Restore puts the code back in place so that the device can try to start up normally again. If you can't get it into DFU mode, then perhaps someone at the Apple store will be able to do this for you and replace it.
Just make an appointment by calling in, and asking for a Genius bar appointment. The benefit of doing it at home will be that it will replace your files from your last backup. So I'd try it at home first, but if you can't get it, certainly take it in for some assistance.
If DFU mode doesn't work, then you may have a buggy / damaged device. But Apple staff can assist / determine that status as well.
Press and hold the Sleep/Wake button for a few seconds until the red "slide to power off" slider appears, and then slide the slider. Press and hold the Sleep/Wake button until the Apple logo appears. Note: Reset your device only if you are unable to restart it.
Press and hold the Sleep/Wake button and the Home button together for at least ten seconds, until the Apple logo appears.
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