On the sets of values of the initial coefficients in class of typically real functions (Q1608604)

Let \(T\) denote the class of holomorphic and typically real functions \(f(z)=z+c_2z^2+c_3z^3+\dots\) in the unit disk \(\mathbb D=\{z:|z|<1\}\) (i.e. \(\operatorname {Im}f(z)\operatorname {Im}z\geq 0\), \(z\in\mathbb D\)). For given fixed \(z_1\in\mathbb D\) and \(w_1\in\{w:w=f(z_1),\;f\in T\}\) the author determines the exact set of values of \(\{c_2\}\) and of \(\{c_3\}\) within the class \(T\). The Carathéodory inequalities for functions with positive real part are used for the proof. All extremal functions are determined.