Extremal problems in some classes of holomorphic functions (Q1320081)

The author considers the family of functions which are analytic in the open unit disk, satisfy \(\text{Re} f(z)>0\) for \(| z |<1\), \(f(0) = 1\), \(f^{(k)}(0)\) is real for \(k = 1,2, \dots\) and \(f^{(k)}(0)\) is fixed for \(k=1,2,\dots,n\). The set of extreme points of this family is determined for each permissible value of \(f^{(k)}(0)\) \((k=1,2,\dots,n)\). That information is used to solve certain extremal problems over the family, including maximizing and minimizing \(f(r)\) and \(f'(r)\) where \(0<r<1\). Similar results are derived for the corresponding family of typically-real functions.