Landau’s Type Inequalities

Let X be a complex Banach space, and let tªT t. 5T t.5F1, tG0. be a strongly continuous contraction semigroup on X.with infinitesimal generator A. This paper proves that 1024 104 5Ax54F 5x535A4x5, 5A2x54F 5x525A4x52, 3 9 5 A3x5 4F1925x5 5A4x53 hold for every xgD A4.. Inequalities are established also for uniformly bounded strongly continuous semigroups, groups, and cosine functions