On value regions of a functional system in the class of typically real functions

E. G. Goluzina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 13, Tome 226 (1996), pp. 69-79

Résumé

Let $T_R$ be the class of functions $$ f(z)=z+\sum^\infty_{n=2}c_nz^n $$ that are regular and typically real in the disk $E=\{z\colon|z|<1\}$. For this class, the region of values of the system $\{f(z_0),f(r)\}$ for $z_0\in E$, $r\in(-1,1)$ is studied. The sets \begin{align*} D_r=\{w\colon w=f(z_0),\ f\in T_R,\ f(r)=a\}\quad&\text{for}\quad-1\le r\le1,\\ \Delta_r=\{(c_2,c_3)\colon f\in T_R,\ -f(-r)=a\}\quad&\text{for}\quad0<r\le1 \end{align*} are found, where $(r(1+r)^{-2},r(1-r)^{-2})$ is an arbitrary fixed number. Bibl. 11 titles.