I think that
the lions win
by the following strategy.
The fast lion moves directly to the centre of the arena, while the slow lion stays on the circumference. When the fast lion reaches the centre, you (the sheep) must be at another point.
After this, the fast lion starts to move outwards, always staying on the radius that connects the centre with the sheep. Thus, in polar coordinates, the fast lion's position changes argument exactly as fast as the sheep's position, and the fast lion has an extra speed component (since its modulus is smaller) which enables it to move outwards towards the sheep.
In this way, the fast lion pushes the sheep further and further to the boundary of the arena. In the one-lion problem, this wouldn't doom the sheep since it can always keep a finite distance away from the lion. But now, the slow lion is lurking on the boundary, waiting for the sheep to get close enough, with the fast lion on the inside, that its measly speed will be enough to reach the sheep. This must happen eventually, since the sheep's modulus approaches 1 (while its argument covers all possible values, it must keep moving to avoid the fast lion) and the slow lion's speed is positive.
This is inspired by
the third argument (one lion, two wolves) in this answer, as well as the fact that the relative circumferential starting points of the two lions don't matter.