You can try the following options and see if any of them work. I would suggest trying them in order, as they range from mild to much more serious. It is better to try the simplest first.
First make sure you are giving the phone enough time to turn on. Make sure you press and hold the power key until the phone vibrates, and then release it. After that it can take a good 10-15 seconds for the phone to turn on.
If that doesn't work, you should then check the battery and charger cord. If the light is still red, the phone may not be charged enough, or charging at all. You might wish to try using a different charger cord and see if this resolves the problem. Or even try using a USB to connect it to a computer to see if it will charge that way. If either of those resolve the problem, you need a new charger cord.
Next, you can see if the memory card is causing an issue. If you have one in your phone, remove it and try starting the phone. If the phone now starts, your SC card is probably damaged. You would then need to replace it.
Last option, which I would definitely not use unless you have tried the others first, you can try to Force the device to restart. No settings or personal data should get deleted during a forced restart.
To Force Restart your phone....
Make sure the phone has charged before doing this.
1. Remove the cover of the memory card slot.
2. Using the tip of a pen or a similar object, press and hold down the OFF button until the device shuts down. (Do not use a pin or other sharp object as that could damage the off button.)
Now try turning off the device again, using the first option listed above.
If this does not work, then I would suggest contacting Sony.