As long as boxing and mixed martial arts exist together, each side will have it's proponents arguing that their sports is superior to the other. Tyson Fury thinks he's the best heavyweight combat athlete in the world and UFC President Dana White thinks his heavyweight champion Cain Velasquez is.
In an interview for Power 105.1, UFC lightheavyweight champion Jon Jones was asked how he would fair in a bout with boxing legend Floyd Mayweather:
"To be honest with you, if it was all hands, it would probably be a pretty competitive match. But that's not fighting, either. You can't be at the bar and be like, alright, no kicking, no taking me down, don't stretch my shirt out. It's not like that. What I do, I consider it to be the true art of fighting. To do everything.
Fighting Floyd Mayweather wouldn't even be right. Nobody would want to see that. James Toney did it about two years ago. He lost in the first round. Don't mess with an MMA fighter. You don't want to do that. It's a bad look."
However, Larry Brown doesn't think a boxing match between the two would even be competive:
I know that Jones is an awesome MMA fighter and that he has a significant size advantage over the 5-foot-8, 147-pound Mayweather, but let’s be real: Mayweather is one of the best boxers in the planet. If we’re talking strictly boxing, Jones is probably considered below average. Regardless of size, Floyd would win that fight.
And don’t even get me started with how it would go in MMA. As we saw with James Toney, the UFC is no place for a boxer. Not unless you’re also competent as a wrestler and schooled in practices like Jiu-Jitsu.