Also, ich würde gebtauchen: 'I hope, you are (all) fine' - Dein Beispiel würde ich eher für Geschäftsangelegenheiten benutzen . . .

Corr: . . . gebrauchen . . .

Im Nachgang fällt mir das Naheliegende ein: 'I hope you are (all) coping alright/well' . . .

'I am fine. I hope you are fine' is sort of stereotypical child talk, what children write in their first letters before they know more vocabulary words. I wouldn't recommend it in any adult context.

But there are lots of other options, like

I hope you are doing well -- not wrong, but 'doing' is a bit colloquial, 'well' is a bit formal
I hope you are well -- rather formal, somewhat ambiguous, since 'well' can mean 'not sick'
I hope you're (doing) all right / okay -- more natural-sounding, casual, slightly worried
I hope things are going all right for you -- also idiomatic, still a bit casual
I hope everything is all right with you -- neutral, perhaps best for a business acquaintance
I hope you weren't (too badly) affected by the storm -- what I would say in this case (-;

I hope you're coping all right -- only if you know they're having problems


Please note the correct spelling of 'all right'; it's often misspelled, but it's definitely two words.

@ hm -- us: A very interesting (if not remarkable) list, indeed. - Thank you.

When I suggested 'coping', I had in mind what spencerpo wrote:

"Möchte jemanden fragen, ob es ihm gut geht (auf den philippinen, nach dem Taifun)."

So, I guess, even to your (high) standards, it's the right choice.

Oh, by the way: LEO has both spellings for alright/all right . . .

Webster's Third also accepts "alright", to which it says "in reputable use, although 'all right'is more common."

Was ich auch schon in einem e-mail gesehen habe:
I hope this finds you well.
Ist das eigentlich korrekt?

Hochschmeiss in eigenem Interesse ; )