Shiryu > Mihawk.
Shiryu was
more deadly than even Magellan, who can do this to Crocodile, Jinbe and Luffy.
What the
same Crocodile does against Mihawk?
Solos him.
Mihawk wasn't even
the most dangerous Shichibukai, it was
Doflamingo.
As for the title, titles are all rumors and people can say other people are better swordsman than Mihawk as we've seen in Zoro novel, there was no officially proven titles, they were discussing who is a better swordsman.
Zoro Novel: ''One marine claims that the strongest swordsman in the world is definitely Mihawk, the other marine claims it's Zoro. They witnessed their sword techniques in marineford and dressrosa. Mihawk cut down gigantic ice block. Zoro cut down gigantic Pica. They start talking about other candidates: Vista, Shanks and Ryuuma. A marine says Vista was able to fight Mihawk equally for a short moment. Ryuma is strong, but he is already dead. According to another marine, Shanks is said to be the rival of Mihawk. He faced Akainu without fear. 2 of the marines have heard that Shanks is a very strong swordsman. But another marine says it's not that Shanks showed his great technique with his sword in Marineford.''
--- Translation by Sandman
Vista had a
competitive fight with Mihawk, which ended in a draw.
"Vista has the skills to compete with Mihawk, one of the most famous swordsmen in the world. He wasn't afraid to confront him..."
--- Vivre Card - One Piece Visual Dictionary
Now why the skills are very important in swordsmanship? Mihawk himself said it.
Mihawk states that no sword is strong
without grace (skill).
Mihawk: ''A sword without grace is no strong sword.''
--- Chapter 0050
Another example is that when Luffy tries to use a sword;
Luffy's performance against Arlong was downgraded, he become even weaker due to lack of sword skills.
And we also know Blackbeard pirates defeated Whitebeard commanders, it's obvious that Blackbeard fought Marco, but we don't know who fought Jozu and Vista. Either way, if Shiryu defeated Jozu or Vista, he would have a better portrayal than Mihawk,
who failed against Jozu as well.
Now we know Shiryu is a better challenge than Vista for Mihawk, but then Shiryu even gets an upgrade with the invisibility power? It's possible that Shiryu already surpassed Mihawk.