Kaplan classes of a certain family of functions

Szymon Ignaciuk, Maciej Parol

Abstract


We give the complete characterization of members of Kaplan classes of products of power functions with all zeros symmetrically distributed in \(\mathbb{T} := \{z \in\mathbb{C} : |z| = 1\}\) and weakly monotonic sequence of powers. In this way we extend Sheil-Small’s theorem. We apply the obtained result to study univalence of antiderivative of these products of power functions.

Keywords


Kaplan classes; univalence; close-to-convex functions; critical points

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References


Goodman, A. W., Univalent functions. Vol. II, Mariner Pub. Co., Inc., Tampa, Florida, 1983.

Ignaciuk, S., Parol, M., Zeros of complex polynomials and Kaplan classes, Anal. Math. 46 (2020), 769–779.

Jahangiri, M., A gap condition for the zeroes of certain polynomials in Kaplan classes K(alpha, beta), Mathematika 34 (1987), 53–63.

Kim, Y. J., Merkes, E. P., On certain convex sets in the space of locally schlicht functions, Trans. Amer. Math. Soc. 196 (1974), 217–224.

Royster, W. C., On the univalence of a certain integral, Michigan Math. J. 12 (1965), 385–387.

Ruscheweyh, S., Convolutions in Geometric Function Theory, Seminaire de Math. Sup. 83, Presses de l’Universite de Montreal, Montreal, 1982.

Sheil-Small, T., Complex Polynomials, Cambridge University Press, Cambridge, 2002.




DOI: http://dx.doi.org/10.17951/a.2020.74.2.31-40
Date of publication: 2020-12-28 17:41:58
Date of submission: 2020-12-27 16:49:42


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