Bat-blog to give this image the credit its due.
Doc was there first. His pulp novels are from "The Doc Savage Magazine" that got its start in 1933... 181 episodes.... ending in 1949. he lived in a fictional building in New York, carried many gadgets of his own invention, was trained in martial arts from the worlds greatest masters, and had a driven "no matter what" code against killing.
Batman got his first print in 1939. This makes him a year older that the spirit. who's first printing was in 1940. The Spirit didn't have the money of the other two, and pretty much made do with humor and chutzpah.
The printing dates of these three also means that the batman comics shared ten years of time with what was coming out of the Doc Savage and Spirit camps... As they were all in the same general field, I do not doubt the artists and writers were aware of each other. If these three creations had not been under separate publishing houses and artists, there is a good argument for making them family.
But really, that's just my wishful thinking. I'd love to see a comic where Batman, Doc Savage, and The Spirit, all in their prime, would compete with each other to solve some sort of mystery. If this is done... please, for the love of all that is nerdy, keep me informed.
(update) its been done, and Phil Noto is the artist. from the picture above... that took a bit to find.