On an estimate of a functional in the class of holomorphic univalent functions

Zbigniew Jerzy Jakubowski; Krystyna Zyskowska

Mathematica Bohemica (1993)

  • Volume: 118, Issue: 3, page 281-296
  • ISSN: 0862-7959

Abstract

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Let S denote the class of functions f ( x ) = z + a 2 z 2 + a 3 z 3 + ... univalent and holomorphic in the unit disc Δ = { z : z < 1 } . In the paper we obtain an estimate of the functional a 3 - c a 2 2 + c a 2 n in the class S for arbitrarily fixed x ∈ 𝐑 and n = 1 , 2 , 3 , ... . 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.

How to cite

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Jakubowski, Zbigniew Jerzy, and Zyskowska, Krystyna. "On an estimate of a functional in the class of holomorphic univalent functions." Mathematica Bohemica 118.3 (1993): 281-296. <http://eudml.org/doc/29081>.

@article{Jakubowski1993,
abstract = {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 = \lbrace z:\left|z\right|<1\rbrace $. 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 $x\in \mathbf \{R\}$ 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.},
author = {Jakubowski, Zbigniew Jerzy, Zyskowska, Krystyna},
journal = {Mathematica Bohemica},
keywords = {estimation of functionals; Koebe function; univalent function; coefficient problem; estimation of functionals; Koebe function},
language = {eng},
number = {3},
pages = {281-296},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On an estimate of a functional in the class of holomorphic univalent functions},
url = {http://eudml.org/doc/29081},
volume = {118},
year = {1993},
}

TY - JOUR
AU - Jakubowski, Zbigniew Jerzy
AU - Zyskowska, Krystyna
TI - On an estimate of a functional in the class of holomorphic univalent functions
JO - Mathematica Bohemica
PY - 1993
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 118
IS - 3
SP - 281
EP - 296
AB - 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 = \lbrace z:\left|z\right|<1\rbrace $. 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 $x\in \mathbf {R}$ 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.
LA - eng
KW - estimation of functionals; Koebe function; univalent function; coefficient problem; estimation of functionals; Koebe function
UR - http://eudml.org/doc/29081
ER -

References

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  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.) (1957) MR0097523
  2. L. Bieberbach, Über die Ҟoeffizienten derjenigen Potenzreihen, welche schlichte Abbildung des Einheitskreises vermitteln, Preuss. Akad. der Wiss. Sitzungsb. 38 (1916), 940-955, Berlin. (1916) 
  3. P. L. Duren, Univalent functions, vol. 259, Grundlehren der math. Wissenschaften, 1983. (1983) Zbl0514.30001MR0708494
  4. M. Fekete G. Szegö, Eine Bemerkung über ungerate schlichte Funktionen, J. London Math. Soc. 8 (1933), no. 30, 85-89. (1933) MR1574865
  5. G. M. Goluzin, Some questions of the theory of univalent functions, Trudy Mat. Inst. Steklov 27 (1949), 51-56. (In Russian.) (1949) MR0042510
  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). (1986) MR0945161
  7. Z. J. Jakubowski A. Szwankowski, On some extremal problem in the class of holomorphic symmetric univalent functions, Commentationes Mathematicæ 29(1990), 195-207. (1990) MR1059124
  8. J. A. Jenkins, On certain coefficients on univalent functions, Princ. Univ. Press, New Jersey, 1960, pp. 159-194. (1960) MR0117345
  9. K. Lowner, 10.1007/BF01448091, Mat. Ann. 89 (1923), 103-121. (1923) MR1512136DOI10.1007/BF01448091
  10. K. Zyskowska, On an estimate of some functional in the class of odd bounded univalent functions, Acta Universitatis Lodziensis 5 (1992), 169-191. (1992) Zbl0822.30018MR1240139

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