The region of values of the system $\{f(z_1),\dots,f(z_n)\}$ in the class of typically real functions.~III

E. G. Goluzina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 23-34

Résumé

The paper studies the region of values $D_{m,n}(T)$ of the system $\{f(z_1),\ldots,f(z_m),f(r_1),\ldots,f(r_n)\}$, where $m\geqslant1$; $n>1$; $z_j$, $j=1,\ldots,m$, are arbitrary fixed points of the disk $U=\{z;|z|1\}$ with $\operatorname{Im}z_j\ne0$, $j=1,2,\dots,m$; $r_j$, $0$, $j=1,2,\dots,n$, are fixed; $f\in T$, and the class $T$ consists of functions $f(z)=z+c_2z^2+\dots$ regular in the disk $U$ and satisfying the condition $\operatorname{Im}f(z)\cdot\operatorname{Im}z>0$ for $\operatorname{Im}z\ne0$, $z\in U$. An algebraic characterization of the set $D_{m,n}(T)$ in terms of nonnegative-definite Hermitian forms is provided, and all the boundary functions are described. As an implication, the region of values of $f(z_1)$ in the subclass of functions $f\in T$ with prescribed values $f(r_j)$ ($j=1,2,3$) is determined. Bibliography: 12 titles.