Verizon iphone 4 showing apple logo then turning off?

Please help me. I bought this iPhone 4 off of ebay. When I got it, it was working fine, but then I noticed in mid use, it would just shut down. I thought it was a one time thing so i didn't sweat it much. I connected it to the charger and turned it on, it was working in perfect order, until I disconnected it, it shut down, which was weird, since the phone read 100% battery. Then after that when I would try to turn the phone on, it would just show the apple logo, and turn off. Now I thought it was a bad battery, so I bought a battery and replaced it. After that the problem persisted. Does anyone know what's wrong with it?

I think it bricked you should connect it to the pc and restoring it or if that doesn't work its best you try to jailbreak it i guess jailbreaking will overide alot of stuff that might b happening i bought and iphone 4 it did the same thing i jailbroke it with greenpois0n and i works fine now