I had black mold on my bedroom ceiling and walls near the bathroom doorway, presumably caused by the moisture from taking showers. I cleaned it once or twice a year with a bleach/water mixture, but it always came back.
Finally I heard that borate would kill mold and prevent it from coming back. It worked! I used Borax detergent, which contains borate, and dissolved some in water in a bucket. I washed the mold from the walls and ceiling using a sponge, then wiped the area down one last time and let it air dry. Letting the borate air dry on the surface is the trick to preventing the mold from coming back. I’ve done this in two rooms in my house and it has worked great. I doubt it would be a lasting solution in a shower or tub that gets used on a regular basis.