Obtainment of the norm in Bergman spaces.
For any holomorphic function on the unit polydisk we consider its restriction to the diagonal, i.e., the function in the unit disc defined by , and prove that the diagonal map maps the space of the polydisk onto the space of the unit disk.
We shall be concerned in this paper with an optimization problem of the form: J(f) → min(max) subject to f ∈ 𝓕 where 𝓕 is some family of complex functions that are analytic in the unit disc. For this problem, the question about its characteristic properties is considered. The possibilities of applications of the results of general optimization theory to such a problem are also examined.