Bakuman was able to get permission to use a number of titles such as One Piece, Naruto, Hunter x Hunter, etc. The creators of the manga also work for Jump. Many of the characters in Bakuman are based on real life mangakas, assistants and editors working in Bakuman.
The number of times it is used is mind boggling and seem odd to me. To be honest while companies pay for product placement, Jump had the perfect opportunity to add its own in an in-house manga/anime?
So, why did Bakuman use Jack instead of Jump, when they could have used it seemingly quite easily?
For example: Gintama has no problem using Jump in its anime.
In the manga translation I read, the magazine was called Shonen Jump, but was changed to Jack in the anime.