Hemp or flax oil are good for the skin. You can put some in his food and also some on the itchy skin. This cured my son's eczema and it also made a huge difference in my older dog's discoid lupus. It's just great for skin issues.
As for food, either feed a raw meat and bone diet (best) or a very, very high-quality kibble with added home-cooked meat and ground vegetables. Good brands to try are Innova (Evo is what we used, it was great, and anything by that company is great), Merrick, Wellness, and other things you can find at those little dog specialty food shops. They're a bit more expensive but for a little dog like a shih tzu it's not bad. Just be sure to add plenty of real food in addition to the kibble. Raw or cooked ground meat, RAW chicken bones, ground vegetables, or even a few bites of whatever you're eating (if it's any kind of vegetables or grains, put it through the blender first to make it more digestible).
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Kadee Sedtal
"I wish I had a dollar for every time I spent a dollar, because then, yahoo! I'd have all my money back." -Jack Handey