Some subordination relations
In this paper \(\gamma\) denotes the unit disc \(| z| <1\) and S the class of functions f(z) analytic and univalent in \(\gamma\), with \(f(0)=0\quad f'(0)=1.\) For a function g(z) analytic in \(\gamma\) with \(g(0)=0,\quad g(z)\prec f(z)\) means that g(z) is subordinate to f(z) in \(\gamma\). The authors show that if \(P_ n(z)=\sum^{n}_{k=1}a_ kz^ k,\quad a_ 1=1,\quad P_{n+1}(z)=P_ n(z)+a_{n+1}z^{n+1},\) and \(P_{n+1}(z)\in S\), then \(P_ n(z/2)\prec P_{n+1}(z)\) for \(n\geq 1\), and the constant \(1/2\) is best possible. Further subordination relations between successive partial sums are also included in the paper.
scientific article; zbMATH DE number 3888007