Based on my understanding, "I hope you are doing well" is usually used to tell a person that the speaker hopes the person has everything going on smoothly. "I hope you are doing good" is, as far as I'm concerned, wrong. "I hope you are fine" is usually used by the speaker to tell a person who is usually mentally or physically hurt.
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AnonymousWhich one is more appropriate?It depends on the situation. It's difficult to think of all the possible situations you might want to know about, so let's restrict it to just a few. Note that it is a common mistake to substitute good for well.