This may be authentic. Unfortunately, I just don’t know this style well enough to say yes or no. If I remember right, however, it seems like this is a style that RR has auctioned off. I’m not 100% sure though. I do know that, at least earlier in his career, Prince had an autograph style that was very, very neat, and different from this.
Here's one from 1988 where he signed with just his first initial. All of the Prince one's I've seen from 1988 just have his first initial. The one you posted I have doubts.