A friend tells me that he has written a not-so-positive memorial piece about someone who died recently, and is looking for a label for the essay. It’s not a “eulogy,” because that is generally considered a positive statement.
I swear I have heard a term for a negative commentary on the life of a recently deceased person, but I sure can’t think of it.
That is what my dictionary offers. I found myself reaching for “execration” or “excoriation”, but then noted the OP’s “not-so-positive”, which might be a smoldering understatement or might actually rule out genuine condemnation.
I’m no linguist, but “eu-” and “dys-” seem to serve as opposites for each other (euphoria, dysphoria), whereas “ben-” and “mal-” are similarly paired up (benevolent, malevolent).
The word eulogy is of Greek origin, so your assumptions are correct. Mal is not a Greek word, but rather based upon the Latin malus (adv.: male).
Of further note: a eulogy does not have to be delivered about a dead person; it is literally just a speech or writing in praise of a person (the original Greek would mean more or less “good words”).
“Dyslogy” is a legit word, and the precise structural antonym to “eulogy”. But it’s very rare. Google NGram shows a ridiculously low but non-zero occurrence rate.