On typically real functions which are generated by a fixed typically real function

Sobczak-Kneć, Magdalena ; Trąbka-Więcław, Katarzyna

Czechoslovak Mathematical Journal, Tome 61 (2011) no. 3, pp. 733-742.

Résumé

Let ${\rm T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\{ z \in \mathbb {C} \colon |z|1 \}$, normalized by $f(0)=f'(0)-1=0$ and such that $\mathop {\rm Im} z \mathop {\rm Im} f(z) \geq 0$ for $z \in \Delta $. In this paper we discuss the class ${\rm T}_g$ defined as \[{\rm T}_g:= \{ \sqrt {f(z)g(z)} \colon f \in {\rm T} \},\quad g \in {\rm T}.\] We determine the sets $\bigcup _{g \in {\rm T}} {\rm T}_g$ and $\bigcap _{g \in {\rm T}} {\rm T}_g$. Moreover, for a fixed $g$, we determine the superdomain of local univalence of ${\rm T}_g$, the radii of local univalence, of starlikeness and of univalence of ${\rm T}_g$.

Détail

DOI : 10.1007/s10587-011-0022-1

Classification : 30C45, 30C55
Keywords: typically real functions; superdomain of local univalence; radius of local univalence; radius of starlikeness; radius of univalence