Press and hold Home and Power button until something appears on the screen. If it's the Apple logo, let go of both buttons. Usually, your iPhone should power up on it's own.
If it's the iTunes logo, put your device into recovery mode and restore it. Here are the instructions: If you can't update or restore your iPhone, iPad, or iPod touch - Apple Support
If nothing happens when you press the buttons, visit the next Apple Retail store, let them have a look at your device.