| Title:
|
On typically real functions which are generated by a fixed typically real function (English) |
| Author:
|
Sobczak-Kneć, Magdalena |
| Author:
|
Trąbka-Więcław, Katarzyna |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
733-742 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 ${\rm T}_g$. (English) |
| Keyword:
|
typically real functions |
| Keyword:
|
superdomain of local univalence |
| Keyword:
|
radius of local univalence |
| Keyword:
|
radius of starlikeness |
| Keyword:
|
radius of univalence |
| MSC:
|
30C45 |
| MSC:
|
30C55 |
| idZBL:
|
Zbl 1249.30045 |
| idMR:
|
MR2853087 |
| DOI:
|
10.1007/s10587-011-0022-1 |
| . |
| Date available:
|
2011-09-22T14:42:21Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141634 |
| . |
| Reference:
|
[1] Golusin, G.: On typically real functions.Mat. Sb., Nov. Ser. 27 (1950), 201-218. MR 0039060 |
| Reference:
|
[2] Goodman, A. W.: Univalent Functions.Mariner Publ. Co., Tampa (1983). Zbl 1041.30501 |
| Reference:
|
[3] Koczan, L., Zaprawa, P.: On typically real functions with $n$-fold symmetry.Ann. Univ. Mariae Curie-Sklodowska, Sect. A, Vol. L II 2 11 (1998), 103-112. Zbl 1010.30019, MR 1728062 |
| Reference:
|
[4] Rogosinski, W. W.: Über positive harmonische Entwicklungen und typischreelle Potenzreihen.Math. Z. 35 (1932), 93-121. MR 1545292, 10.1007/BF01186552 |
| Reference:
|
[5] Todorov, P. G.: The radii of starlikeness and convexity of order alpha of typically real functions.Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), 93-106. MR 0698840, 10.5186/aasfm.1983.0824 |
| . |
