Doesn't power on, blinks the logo when connected to current or laptop

My iPhone 4s logo comes off and on when plugged to light and laptop, when you hold the home and power button it shows the connect to iTunes symbol but turns off immediately it's disconnected from power source hence I think it wont format/restore properly, I have changed the battery (which is ok as it powers others), checked and made sure the battery connectors and terminals are in order yet the problem persist. Please what can I do?

Some one please help with my question. Thanks