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Put some compound on a microfiber towel, wrap the towel around a finger or two, rub like heck. Repeat with a polish and refine as necessary (might not even be needed on a white car). Should take you <5 minutes.
My girlfriend does this all the time to ours and it drives me nuts. Most people treat the doors on their cars like barn doors. In all the years I've owned cars this has never been a problem because I make it a point not to touch the door handle cups everytime I open a door. I might have my ExPel guy cover this area on my next Lexus when I get it because the girlfreind never listens when I mention it to her.
As the others have suggested, some polishing compound and a microfiber will do the trick. Be sure to throw some wax/sealant over it once you're done, that way it'll be a little more protected. You can also put small patches of clear, protective film in that area. Did it with mine and am very pleased.
I’ve always had good luck with Meguiars Polishing compound on my white car door handles. After I put Cquartz ceramic coating on it, I’ve not had any issues since.
Toyota and Lexus Join Mille Miglia For The First Time
Slideshow: A five-car lineup spanning more than five decades of Toyota performance and engineering will tackle one of Italy's most celebrated automotive routes.