"Jax is the best duelist"
not even close

not even close

In what alternate reality would Jax lose in a duel to Riven, Vi, Mundo or Shyvana? Really? Are we talking level 3 or something?
On full builds, it would come to skill matchups and situations, but the only ones who could fight a Jax would be Nasus, Trynd, Malph and Ryze. And even then, it's hard to say who'd win.
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I've been late game Jax vs late game Shyvana quite a few times. And it isn't pretty for Shyv.
The only people jax would actually lose to in a duel would be MAYBE nasus or mundo. Most others are not too difficult.
If you lose against riven past level 9 and with equal equip then you are a bronze 5 jax.
His ult passive is absolutely insane. He can outduel most of these with ease.
Jax beats all of these, although Tryndamere, Olaf, Malphite, Ryze, and Fiora are tough. These can easily win if Jax isn't toe to toe in gold and items with them. Nasus is entirely depending on the situation with how many stacks he has. Usually amounts say Jax wins though (450-650).
Garen beats Jax, but Garen is also not actually a duelist.
Malph?
You forgot 

"duelist"

Best duelist? Maybe.
What Jax has is the best DPS/Tankiness mix. He is always going to win when the fight comes to trading blows. It's possible for (most of) the other champions listed above to defeat a Jax, but they need to get the outplays
He can beat Fiora if she parries first.
[{quoted}](name=I Win Haha,realm=NA,application-id=3ErqAdtq,discussion-id=1qKQQmYW,comment-id=,timestamp=2015-11-04T21:12:37.945+0000)
not even close
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Shyvana is easy. Gnar is easy. Darius? Super easy. Trundle I'm not sure on. Tryndamere is a skill match up and depends on build IMO. Olaf frankly doesn't have the tankiness or damage to fight Jax, especially since Jax can mitigate a significant portion of Olaf's DPS (even if he can't stun him). Nasus can indeed duel Jax, and pretty easily too. Garen? Only if Jax is the Villain, and maybe not even then. Malphite perhaps. A well-played Ryze is probably really hard, because of his near perma root. Fiora is a skill match up (fun lane, played it as Jax recently). Vi? lol. Mundo? Doubtful. Riven? Pffffffthahahhahahahah!
Fiora and jax i think are about even. it would come down to who did better in the use of their dodge/parry ability. Whoever used it first is gonna lose.