Several years ago, I posted my own personal chrono data on various pistol caliber carbines on this board. In the 9mm I started with a KelTec 9mm and went up to a Colt AR15 in 9mm with a 16" barrel. I won't post the link: I have done so many times over the years, but the data supports what has already been posted. No matter what bullet or powder you use, there is only a small increase in velocity when using a longer barrel. To me, it was interesting that you can get close to the same performance out of a very tiny 9mm handgun as you can a service sized handgun.
I can't think of anything I would actually use a 9mm carbine for, other than fun. I have other guns that do everything better than a 9mm carbine for defensive purposes. But, the 9mm carbines are a LOT of fun. If you lived in a state that permitted private ownership of NFA weapons, you could have a 9mm SBR. I have a Colt Lightweight Sporter (9mm AR15) that I registered as a short barreled rifle. I have two uppers for it (in addition to the factory 16" upper): one is a flat top with a 10 1/2" barrel with a KAC RASII rail system, Aimpoint ML2 in a LaRue mount and a Scully stock. The other upper is an LRM M169 suppressed upper (
http://www.lrmfirearms.com/pages/863787/index.htm ). These are a blast. The shorter barrels are no real compromise in 9mm. 9mm factory ammo is real cheap. I have a bunch of steel targets and I don't have to worry about damaging them with the 9mm. I handload 9mm and like most handgun rounds it is very simple to handload since you can use carbine dies and a progressive loader. I load all my own subsonic stuff. These carbines are also terriffic for stuff like night jackrabbit shooting since you can drive around off road and the tiny little carbine is easy to store and take in and out of the truck as well as easy to fire out the windows.
Right now (this may change in an hour), my 9mm carbine is my faviorite toy. Mine is sort of a clone of the Colt 9mm SMG:
http://www.chucktaylorasaa.com/coltm635.html
Even though you only get a slight increase in velocity with the longer barrel, this would allow you to shoot a heavier bullet out of a carbine at the same velocity as a lighter bullet out of a handgun: for example you could shoot a 147 grain HP at the same velocity out of a carbine as you could shoot a 124 or maybe 115 out of a handgun. Heavier is always good if you don't lose any velocity in the process.
Getting further away from the original question: with revolver cartridges like the .357 and the .45 Colt, the difference between a handgun and a carbine is HUGE. And with these cartridges you can really get into using slower burning powders to optimize the cartridge for use in a carbine.