Pretty easy to just calculate it.
Fire 3 does 62.85 pot/sec
Fire 1 does 60 pot/sec

So if you're just shooting off one of those spells continuously fire 3 wins by a small margin (assuming you've got AFIII and it has the same modifier on both).
But that's obviously not the case. You have to regen mana, something that takes the same amount of time with either skill. Which means it's a higher % of time when doing F3. You also start off the burn phase with a reduced-time F3 in either case. With F1 you're getting free casts of F3.

What's the ratio of mana costs between f1 and f3? I'm pretty sure it's about the same per potency?

I think you might make the case that before you get Firestarter F3 is better. But with it, I think you're much more mana efficient, so your burn phase lasts longer keeping you in the high DPS state longer. That's better for sustained DPS.