Why is the Apple logo blinking on my iPhone 4?

My iPhone 4 had around 50% battery left. The phone shut down and now I can not restart it. I have tried holding down the power button and home screen multiple times. I have also plugged it in to my computer, which will not recognize it, and left it in the wall charger for about a half an hour. I have not dropped the phone and it has not been exposed to water.

I recently went to the Apple store and got the phone replaced becasue the home button was not working and there were various cracks throughout the phone. Is it possible that there was a problem with the refurbished phone they gave me?

Thanks