Regions of values of the systems $\{f(z_1),f(z_2),f'(z_2)\}$ and $\{f(z_1),f'(z_1),f''(z_1)\}$ on the class of typically real functions

E. G. Goluzina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 18, Tome 286 (2002), pp. 48-61

Résumé

Let $T$ be the class of functions $f(z)=z+a_2z^2+\dots$ that are regular in the unit disk and satisfy the condition $\operatorname{Im}f(z)\operatorname{Im}z>0$ for $\operatorname{Im}\ne0$, and let $z_1$ and $z_2$ be any distinct fixed points in the disk $|z|1$. For the systems of functionals mentioned in the title, the regions of values on $T$ are studied. As a corollary, the regions of values of $f'(z_2)$ and $f''(z_1)$ on the subclasses of functions in $T$ with fixed values $f(z_1),f(z_2)$ and $f(z_1),f'(z_1)$, respectively, are found.