An estimate of the Rasmussen invariant for links and the determination for certain links

Improving the slice-Bennequin inequality shown by Rudolph, we estimate some knot or link invariants, especially the knot invariant defined by Ozsváth and Szabó and the Rasmussen invariant for links introduced by Beliakova and Wehrli. Our argument implies a combinatorial proof of the slice-Bennequin inequality for links. Furthermore we determine such invariants for negative links and certain pretzel knots.