| Title:
|
On an estimate of a functional in the class of holomorphic univalent functions (English) |
| Author:
|
Jakubowski, Zbigniew Jerzy |
| Author:
|
Zyskowska, Krystyna |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
118 |
| Issue:
|
3 |
| Year:
|
1993 |
| Pages:
|
281-296 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $S$ denote the class of functions $f(x)=z+a_2z^2+a_3z^3+\ldots$ univalent and holomorphic in the unit disc $\Delta = \{z:\left|z\right|<1\}$. In the paper we obtain an estimate of the functional $\left|a_3-ca^2_2\right|+c\left|a_2\right|^n$ in the class $S$ for arbitrarily fixed $x\in \bold R$ and $n=1,2,3,\ldots$. Hence, for some special values of the parameters, we obtain estimates of several interesting functionals and numerous applications. A few open problems of a similar type are also formulated. (English) |
| Keyword:
|
estimation of functionals |
| Keyword:
|
Koebe function |
| Keyword:
|
univalent function |
| Keyword:
|
coefficient problem |
| MSC:
|
30C50 |
| idZBL:
|
Zbl 0783.30015 |
| idMR:
|
MR1239123 |
| DOI:
|
10.21136/MB.1993.125927 |
| . |
| Date available:
|
2009-09-24T21:00:15Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/125927 |
| . |
| Reference:
|
[1] I. E. Bazilevič: Domains of principal coefficients of bounded univalent functions of p-tuple symmetry.Mat. Sb. 4З (1957), no. 4, 409-428. (In Russian.) MR 0097523 |
| Reference:
|
[2] L. Bieberbach: Über die Ҟoeffizienten derjenigen Potenzreihen, welche schlichte Abbildung des Einheitskreises vermitteln.Preuss. Akad. der Wiss. Sitzungsb. 38 (1916), 940-955, Berlin. |
| Reference:
|
[3] P. L. Duren: Univalent functions.vol. 259, Grundlehren der math. Wissenschaften, 1983. Zbl 0514.30001, MR 0708494 |
| Reference:
|
[4] M. Fekete G. Szegö: Eine Bemerkung über ungerate schlichte Funktionen.J. London Math. Soc. 8 (1933), no. 30, 85-89. MR 1574865 |
| Reference:
|
[5] G. M. Goluzin: Some questions of the theory of univalent functions.Trudy Mat. Inst. Steklov 27 (1949), 51-56. (In Russian.) MR 0042510 |
| Reference:
|
[6] Z. J. Jakubowski W. Majchrzak A. Szwankowski: On some extremal problem in the class S of functions holomorphic and univalent in the unit disc.Ann. UMCS 40 (1986), 63-75 (1988). MR 0945161 |
| Reference:
|
[7] Z. J. Jakubowski A. Szwankowski: On some extremal problem in the class of holomorphic symmetric univalent functions.Commentationes Mathematicæ 29(1990), 195-207. MR 1059124 |
| Reference:
|
[8] J. A. Jenkins: On certain coefficients on univalent functions.Princ. Univ. Press, New Jersey, 1960, pp. 159-194. MR 0117345 |
| Reference:
|
[9] K. Lowner: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises.Mat. Ann. 89 (1923), 103-121. MR 1512136, 10.1007/BF01448091 |
| Reference:
|
[10] K. Zyskowska: On an estimate of some functional in the class of odd bounded univalent functions.Acta Universitatis Lodziensis 5 (1992), 169-191. Zbl 0822.30018, MR 1240139 |
| . |
