b3.5 is the best moves for white.
It is not necessary for white to play h9.
However, if white does not follow a strict sequece, white looses the entire right side.

1 W: b3.5
2 B: h9
3 W: i8 (white must play i8 or g9.5. If white plays elsewhere, black follows with d9 and the entire white right side perishes)
4 B: h6 (threats the white group at lower right)
5 W: g9.5 atari (trying a ko fight. If white does not play g9.5, the entire right side white groups will die.)
6 B: f9.5 eats white g9.5.
7 W: plays elsewhere. a5 for example.
8 B: d9 atari.
9 W: g9.5 atary on the black group of 3. (take out black on f9.5)
10 B: e8 (takes out the white group of 2) (the two white stones are dead doesn't matter how white plays)
11 W: i9 takes out the black group of 3, ends the battle.