The authors prove an inequality for sums, which generalizes both Landau's sharpening of Carlson's inequality and the corresponding complementary result by Levin and Ste\v{c}kin. The inequality is optimal, in the sense that necessary and sufficient conditions on the parameters for which the inequality holds are given. In some cases, sharp constants are obtained, also in situations not covered by the classical results.
In this paper we study the effective behavior of composites with a nonlinear material behavior and a special type of microstructure. The microstructure we study is obtained by reiteration, i.e., introducing several local scales. We derive bounds on the effective shear properties and prove that the bounds are sharp when the number of scales tends to infinity.