Title
Which is the better starting item on 
?
I've been going flask a lot because it seems to give me the mana for harass and helps me save gold for large rod.
Though I do know Doran's ring give more damage.
Edit:
Is the 15 AP significant damage? That's like 10 damage on her q. Compared to Doran's sword its seems much weaker in DPS since auto's have a lower "cool down".
take flask you dont need that extra damage considering your charm increases the damage of your q
Ring if you plan of actually fighting your opponent, flask if you just want to survive and farm.
I remember thinking that starting AP was fantastic and nice. Then I grabbed flask on a whim one game. Assuming you are going standard mid, the trick is to outlast them. Most mids will either save their skills to grab ungrabbable cs or spam it in your face to deny you cs/kill you.
TM;DU? If its mana based, grab flask and outlast em. If its energy or cooldown based, grab ring and try not to die.
I have a different build from most, I take two faerie charms and 3 potions as starting items, this gives me decent regen and the ability to stay in lane and harass and farm until I can finish Chalice of Harmony.
The 15 AP helps add up later. Also that extra AP later can mean the difference of clearing out the minion wave or not. The usefulness of ring is its HP/AP, but ALSO its mana regen and kill mana regen. This helps you sustain through lane and also be able to wave clear a lot easier if you're able to last hit well. In general, though, your choice between flask/doran's comes down to whether you're in a tough lane matchup. If you're against someone who will harass you alot or be able to bully you out, you want to go flask so you can regen more (this also saves you money of pots in the overall laning phase when you would be getting pots). If you aren't going to need the sustain, though, you might as well go the ring for the extra damage.
If you don't care for the differences, though, just go with whichever you like as it doesn't offer TOO much of a difference, just are more useful in certain scenarios.
I prefer dorans on ahri. 1. Because she has built in spell vamp, hitting the entire wave with orb brings significant returns if you need them. 2. I haven't done the numbers but to me you get more mana regen from the dorans ring due to its passive.
Flask becomes pretty useless quickly. A lot of mid laners can easily all in you once level 6 is reached and flask does not protect you from burst.
Also as someone said ring helps with wave clear which also quickly becomes important.
