Multi-Dimensional Landau Extremum Problems

Edmund Landau [5] initiated the following extremum problem: The sharp inequality between the supremum-norms of derivatives of twice differentiable functions f such that
‖f'‖²≤4‖f‖ ‖f"‖                                                 (+)
holds with norm referring to the space C[0,∞]. In this work we extend Landau’s results and establish new multi-dimensional inequalities of Landau type for semigroups and cosine functions.