Previous |  Up |  Next

Article

Keywords:
Hadamard product; typically real functions; class of type $A_\alpha$
Summary:
Let $\Cal A$ denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F(0)=0, F'(0)=1$, whereas $A\subset \Cal A$ is an arbitrarily fixed subset. In this paper various properties of the classes $A_\alpha, \alpha \in C \{-1,-\frac{1}{2},\ldots\}$, of functions of the form $f=F*k_\alpha$ are studied, where $F\in .A$, $k_\alpha(z)=k(z,\alpha)=z+\frac{1}{1+\alpha}z^2+\ldots + \frac{1}{1+(n-1)\alpha}z^n+\ldots$, and $F*k_\alpha$ denotes the Hadamard product of the functions $F$ and $k_\alpha$. Some special cases of the set $A$ were considered by other authors (see, for example, [15],[6],[3]).
References:
[1] L. Brickman D. R. Wilken: Support points of the set of univalent functions. Proc. Amer. Math. Soc. 42 (1974), 523-528. DOI 10.1090/S0002-9939-1974-0328057-1 | MR 0328057
[2] D. M. Campbell V. Singh: Valence properties of solution of differential functions. Pac. J. Math., v. 84, No. í (1979), 29-33. DOI 10.2140/pjm.1979.84.29 | MR 0559624
[3] P. N. Chichra: New subclasses of the class of close - to - convex functions. PAMS, v. 62,. No. 1 (1977), 37-43. DOI 10.1090/S0002-9939-1977-0425097-1 | MR 0425097 | Zbl 0355.30013
[4] A. W. Goodman: Univalent functions. v. 1, 2. Mariner Publishing Company, 1983. Zbl 1041.30501
[5] I. S. Jack: Functions staгlike and convex of order $\alpha$. J. London Math. Soc., v. 2, No. 3 3 (1971), 469-474. DOI 10.1112/jlms/s2-3.3.469 | MR 0281897
[6] Z. J. Jakubowski: On some classes of analytic functions. Complex Analysis. Sofia (1989), 241-249. Proc. of the IXth Instructional conference on the theory of extremal problems, Sielpia (1987), 59-78. MR 1127640
[7] J. Krzyż Z. Lewandowski: On the integral of univalent functions. Bull. Acad. Polon. Sci., 11, 7 (1963), 447-448. MR 0153830
[8] Z. Lewandowski S. Miller E. Zlotkiewicz: Generating functions for some classes of univalent functions. Proc. Amer. Math. Soc., 56 (1976), 111-117. DOI 10.1090/S0002-9939-1976-0399438-7 | MR 0399438
[9] S. Miller: Differential inequalities and Caгathéodory functions. Bull. of Amer. Math. Soc. v. 81, No. 1 (1975), 79-81. DOI 10.1090/S0002-9904-1975-13643-3 | MR 0355056
[10] N. N. Pascu: Јanowski alpha-starlike-convex functions. Studia Univ. Babes-Bolyai, Mathematica (1976), 23-27. MR 0396929
[11] M. S. Robertson: Extremal problems for analytic functions with positive real part and applications. Tгans. Amer. Math. Soc., 106 (1963), 236-253. DOI 10.1090/S0002-9947-1963-0142756-3 | MR 0142756
[12] W. W. Rogosiński: Übeг positive harmonische Entwicklungen und typisch - reel Potenzreihen. Math. Z., 35 (1932), 93-121. DOI 10.1007/BF01186552 | MR 1545292
[13] S. Ruscheweyh: Convolutions in geometric function theory. Press Univ. Montreal, 1982. MR 0674296 | Zbl 0499.30001
[14] S. Schober: Univalent functions - selected topics. Lecture Notes in Math., 1975. Zbl 0306.30018
[15] K. Skalska: Certain subclasses of the class of typically real functions. Ann. Polon. Math., 38 (1980), 141-152. MR 0599238 | Zbl 0465.30012
[16] O. Toeplitz: Die linearen vollkommen Räume der Funktionentheorie. Comment. Math. Helv., 23 (1949), 222- 242. DOI 10.1007/BF02565600 | MR 0032952
Partner of
EuDML logo