Parts within the powersupply board will run very hot, OK when the set shuts down are there any blinking lights on the front of the set?
The chevron should be blinking long and short blinks ?so is it one long one short blink?
OK your not going to like what I am going to say about this issue but its a very common one particularly with sharp sets, there is a problem with the backlighting its bad the reason why it stays up in eco mode is because the backlights are "backed off" so are able to run this does not always happen and in most cases the set will stay off blinking like it is, sorry to say but its not an option to replace the backlights in an LCD\LED set due to costs, they are considered integral to the panel so just like the panel they are not replaced. One long\one short means bad backlighting
I am sorry for bad news and I know this was not what you wanted to hear however I do need to be rated positively if I am to receive any credit from the site, rating bar of stars is at the top of the page thank you.
If you need any more help please let me know sorry for dismal news you should complain to sharp via the site www.sharpusa.com
David I am sorry I don't have better news
Please remember to complain to sharp via the website I have know manuf to help particular if the set is under 3 years old