By all contemporary accounts I've read, the Bearcat could fly rings around any other prop fighter, theirs, ours, whoever. It absolutely PASTED some late model 51s and 47s. IMS, this was reluctantly confirmed by some AAC pilots later. Interesting thing about the Hellcat...while not nearly as sexy as the Corsair (let alone the Bearcat or Tigercat), it had a significantly larger effect on the course of air combat, and I want to say (with no reference) that it actually had a higher kill rate, supposedly down to the fact that it was relatively easy to fly and maintained some turning ability, rather than the pure energy fighting required of the more highly wing-loaded late war fighters. Oh, and was armored like an IL-2 (ok, not quite). The FM-2s (basically a stripped down Wildcat with more power) that were flown off the jeep carriers deserve a mention here, too.
Other names the Corsair earned: "Bent-wing bastard", "Hose-nose", and "The Ensign Eliminator".
Don't get me wrong, I think it's a beautiful thing, and its performance in the A2G role in the late Pacific Theater (and esp. in Korea) can hardly be questioned. But as a carrier-based air superiority fighter, let's be sanguine and say that it "had a few teething problems".