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Keywords:
estimation of functionals; Koebe function; univalent function; coefficient problem
estimation of functionals; Koebe function; univalent function; coefficient problem
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.
References:
[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
[2] L. Bieberbach: Über die Ҟoeffizienten derjenigen Potenzreihen, welche schlichte Abbildung des Einheitskreises vermitteln. Preuss. Akad. der Wiss. Sitzungsb. 38 (1916), 940-955, Berlin.
[3] P. L. Duren: Univalent functions. vol. 259, Grundlehren der math. Wissenschaften, 1983. MR 0708494 | Zbl 0514.30001
[4] M. Fekete G. Szegö: Eine Bemerkung über ungerate schlichte Funktionen. J. London Math. Soc. 8 (1933), no. 30, 85-89. MR 1574865
[5] G. M. Goluzin: Some questions of the theory of univalent functions. Trudy Mat. Inst. Steklov 27 (1949), 51-56. (In Russian.) MR 0042510
[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
[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
[8] J. A. Jenkins: On certain coefficients on univalent functions. Princ. Univ. Press, New Jersey, 1960, pp. 159-194. MR 0117345
[9] K. Lowner: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. Mat. Ann. 89 (1923), 103-121. DOI 10.1007/BF01448091 | MR 1512136
[10] K. Zyskowska: On an estimate of some functional in the class of odd bounded univalent functions. Acta Universitatis Lodziensis 5 (1992), 169-191. MR 1240139 | Zbl 0822.30018
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