I bought this at a fair price as a vintage studio picture. It's signed in the back by a fan saying it arrived in April 1927. It's 5 x 7.
I've seen a couple others. One an 8x11 unsigned.
Another signed "Sincerely Yours" on top, "Louise Brooks" below.
The one I have just says "sincerely" with her name below.
I've studied a number of signatures on these and different pictures. All are slightly different in terms of lettering and spacing. No two are exactly alike. So I doubt they are stamped. Most likely, secretarial. However, they are really similar to many authenticated Brooks signatures.
I would be curious about further information and opinions. It's a nice piece of vintage memorabilia regardless. 