Classes of analytic functions (Q1081719)

Let \(L_ n(E,t)\) (resp. \(C_ n(E,t))\) be the class of functions \[ F(z):=(z+t)^ n(1+a_ 1z+a_ 2z^ 2+...) \] holomorphic in the disc \(E=\{| z| <1\}\) such that \(\sum^{\infty}_{k=1}\delta_{nk}(t)| a_ k| \leq 1\) (resp. Re \(F^{(n)}(z)>0\) in E), where \[ \delta_{nk}(t):=\max_{| z| \leq 1}| \frac{1}{n!}(\frac{\partial}{\partial z})^ n((z+t)^ nz^ k)|. \] The authors are studying properties of the two classes. For some special values of n and t the two classes were earlier studied by various authors. In particular, the class \(L_ 1(E,0)\) was investigated by Lewandowski and Zmorovich, \(C_ 1(E,0)\) is the Carathéodory class.