Typically real functions and typically real derivatives (Q1099285)

Let D denote the unit disk \(\{\) \(z: | z| <1\}\). W. W. Rogosinski defined the class T of typically real functions \(f(z)=z+a_ 2z^ 2+..\). regular in D and such that Im z\(=0\) if, and only if, Im f(z)\(=0\). The author modifies this definition to remove the normalization conditions on f and, motivated by the fact that \(f(z)=z+z^ 3\in T\) while f is not univalent, he gives sufficient conditions in terms of typically real derivatives which ensure that functions are univalent. In particular, he defines \(T'=\{f\in T:\) f' is also typically real in \(D\}\) and shows that if \(f\in T'\) then f is univalent in D. Furthermore, let E denote the class of functions \(f(z)=z+a_ 2z^ 2+..\). such that \(f^{(n)}\) is univalent in D for \(n=0,1,2,..\). and ER those functions \(f(z)=z+a_ 2z^ 2+..\). in E with \(a_ n\) real for \(n=2,3,... \). \(\overline{ER}\) denotes the class of functions which are uniform limits on compact subsets of D of sequences in ER, and ERP those functions in ER such that \(a_ n>0\) for \(n=2,3,... \). The author shows that \(f\in ER\) if, and only if, \(f^{(n)}\) is typically real in D for \(n=0,1,2,..\). and includes various results for the classes ER, \(\overline{ER}\) and \(\overline{ER}\setminus ER\), together with a proof that ERP is convex.