Yes, the artist's name appears on the originals and the reproductions on the front of the tray to the right of the girl. This signature is on all the paintings he created for Coke.
Hamilton King was an artist who painted Coca-Cola girls for calendars from 1910 through 1913. Trays with the girls were also produced. This is the 1910 tray, which was reproduced in 1971 and later in the 1980's. These both had less detail than the original. The edge of the tray says Copyright 1909.