The film was originally released as simply Star Wars in 1977 and was later renamed as Star Wars Episode IV: A New Hope when it was re-released in 1981. However, besides the opening title, the "A New Hope" subtitle wouldn't be shown in the logo until the VHS re-print in 2000.
In 1997, the film received a re-release (along with episodes V and VI) called "Special Edition", in which some of the scenes and special-effects were altered using CGI, among other changes.
Original logo for promotions (1977)
Logo with "Episode IV: A New Hope" subtitle
Logo with "A New Hope" subtitle (seen the 2015 Chapter Sampler)
Logo with "A New Hope" subtitle #2 (used in the 2019 Blu-ray cover)
Alternate variant, which uses the same style of episodes I, II, III logos