There is a rumor (a myth ?myth?) in holomorphic dynamics about a rivality rivalry between Fatou and Julia competing for the 1918 Grand Prize of the french science academy. In fact Fatou never submitted a memoir for the Prize.
What we now call the Julia set was first defined in a note in Comptes-rendus Comptes Rendus by Montel, and is the starting point for the work of Fatou and Julia. Fatou published a few notes on the subject in the Comptes-rendus Comptes Rendus before Julia, who started a quarrel, claiming the priority for the results. Fatou didn't try to fight back, apparently because he didn't bother, and the Academia fullfilled the wish of Julia. This may be the source of the rumor.
You may also know the Parceval Parseval formula about the relationship between the square of the L2 norm of a function and its Fourier coefficients. This formula is due to Fatou. H. Lebesgue did the case of a bounded function, and Fatou extended it to any square integrable functions. This made a big impression on Lebesgue.
This is exposed in a recent book by Michele Audin entitled "Fatou, Julia, Montel (...)"