Extremal Problems for Nonvanishing Functions in Bergman Spaces [chapter]

Dov Aharonov, Catherine Bénéteau, Dmitry Khavinson, Harold Shapiro
Operator Theory: Advances and Applications  
In this paper, we study general extremal problems for non-vanishing functions in Bergman spaces. We show the existence and uniqueness of solutions to a wide class of such problems. In addition, we prove certain regularity results: the extremal functions in the problems considered must be in a Hardy space, and in fact must be bounded. We conjecture what the exact form of the extremal function is. Finally, we discuss the specific problem of minimizing the norm of non-vanishing Bergman functions whose first two Taylor coefficients are given.
doi:10.1007/3-7643-7340-7_5 fatcat:pwzy7mdbjncrrgpezo6s3hfsxi