On properties of non-Carathéodory functions (Q1201502)

Let \(N\) be the class of all functions that are analytic in the unit disk \(E=\{z: | z|<1\}\) and equal to 1 at \(z=0\). The author obtained the following theorem: Theorem. Let \(p(z)\in N\) and suppose that there exists a point \(z_ 0\in E\) such that \(\text{Re }p(z)>0\) for \(| z|<| z_ 0|\), \(\text{Re }p(z_ 0)=0\) and \(p(z_ 0)\neq 0\). Then we have \({z_ 0 p'(z_ 0)\over p(z_ 0)}=ik\), where \(k\) is real and \(| k|\geq 1\).