Logarithmic convexity results for holomorphic semigroups

The classical logarithmic convexity inequality, for solutions of u 1 = -Au with A a self adjoint operator on Hubert space, yield that u is small at intermediate times, 0 < t ^ T, provided that u is small at T and bounded at 0. Use of the Carleman inequality for analytic functions allows one to easily generalize this result to the case of operators A which are generators of holomorphic semigroups on Banach space.