On an extremal problem

Krystyna Zyskowska

Mathematica Bohemica (1995)

  • Volume: 120, Issue: 2, page 113-124
  • ISSN: 0862-7959

Abstract

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Let S denote the class of functions f ( z ) = 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 a sharp estimate of the functional | a 3 - α a 2 2 | + α | a 2 | 2 in the class S for an arbitrary α ∈ ℝ .

How to cite

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Zyskowska, Krystyna. "On an extremal problem." Mathematica Bohemica 120.2 (1995): 113-124. <http://eudml.org/doc/247795>.

@article{Zyskowska1995,
abstract = {Let $S$ denote the class of functions $f(z) = z + a_2z^2 + a_3z^3 + \ldots $ univalent and holomorphic in the unit disc $\varDelta = \lbrace z |z| < 1\rbrace $. In the paper we obtain a sharp estimate of the functional $|a_3 - \alpha a^2_2| + \alpha |a_2|^2$ in the class $S$ for an arbitrary $\alpha \in \mathbb \{R\}$.},
author = {Zyskowska, Krystyna},
journal = {Mathematica Bohemica},
keywords = {coefficient problems; Koebe function; univalent function; coefficient problems; Koebe function},
language = {eng},
number = {2},
pages = {113-124},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On an extremal problem},
url = {http://eudml.org/doc/247795},
volume = {120},
year = {1995},
}

TY - JOUR
AU - Zyskowska, Krystyna
TI - On an extremal problem
JO - Mathematica Bohemica
PY - 1995
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 120
IS - 2
SP - 113
EP - 124
AB - Let $S$ denote the class of functions $f(z) = z + a_2z^2 + a_3z^3 + \ldots $ univalent and holomorphic in the unit disc $\varDelta = \lbrace z |z| < 1\rbrace $. In the paper we obtain a sharp estimate of the functional $|a_3 - \alpha a^2_2| + \alpha |a_2|^2$ in the class $S$ for an arbitrary $\alpha \in \mathbb {R}$.
LA - eng
KW - coefficient problems; Koebe function; univalent function; coefficient problems; Koebe function
UR - http://eudml.org/doc/247795
ER -

References

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  1. L. Bieberbach, Über die Koeffizienten derjenigen Potenzreihen, welche schlichte Abbildung des Einheitskreises vermitteln, Preuss. Akad. deг Wiss. Sitzungsb. 38 (1916), 940-955. Berlin. (1916) 
  2. G. M. Goluzin, Some questions of the theory of univalent functions, Trudy Mat. Inst. Steklov 27 (1949), 51-56. (In Russian.) (1949) MR0042510
  3. Z. J. Jakubowski K. Zyskowska, On an estimate of some functional in the class of holomorphic univalent functions, Mathematica Bohemica 118 (1993), 281-296. (1993) MR1239123
  4. J. A. Jenkins, On certain coefficients on univalent functions, Princ. Univ. Press, New Jeгsey, 1960, pp. 159-194. (1960) MR0117345
  5. W. Ma D. Minda, Uniformly convex functions II, To appear in Ann. Polon. Math. Zbl0792.30008MR1244399
  6. A. C. Schaeffer D. C. Spencer, Coefficient regions for schlicht functions, Amer. Math. Soc., Colloq. Publ. 35 (1950), 36-37. (1950) MR0037908

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