Abstract
We continue research on problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in \(\mathbb {T}:=\{z\in \mathbb {C}:|z|=1\}\). We found minimal disc containing all images of \(\mathbb {D}:=\{z\in \mathbb {C}: |z|<1\}\) and maximal disc contained in all images of \(\mathbb {D}\) through polynomials of degree 5. Moreover, we determine the extremal functions for both problems. Furthermore, we state the conjecture concerning polynomials of higher odd degrees.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Code Availability
Not applicable
References
Bieberbach, L.: Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. Sitzungsber. Preuss. Akad. Wiss. Phys-Math. Kl. 940-955 (1916)
Dmitrishin, D., Dyakonov, K., Stokolos, A.: Univalent polynomials and Koebe’s one-quarter theorem. Anal. Math. Phys. 9(3), 991–1004 (2019). https://doi.org/10.1007/s13324-019-00305-x
Dmitrishin, D., Smorodin, A., Stokolos, A.: An extremal problem for polynomials. Appl. and Comput. Harmonic Anal. 56, 283–305 (2022). https://doi.org/10.1016/j.acha.2021.08.008
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, 769–779 (2020). https://doi.org/10.1007/s10476-020-0044-8
Ignaciuk, S., Parol, M.: Kaplan classes of a certain family of functions. Annal. Univ. Mariae Curie-Sklodowska, Sec. A – Math 74(2), 31–40 (2020). https://doi.org/10.17951/a.2020.74.2.31-40
Ignaciuk, S., Parol, M.: A gap condition for the zeros of a certain class of finite products, Proc Rom Acad Ser A. 22(1), 19-24 (2021). https://acad.ro/sectii2002/proceedings/doc2021-1/03-Ignaciuk_Parol.pdf
Ignaciuk, S., Parol, M.: On the Koebe Quarter Theorem for certain polynomials. Anal. Math. Phys. 11(2), 1–12 (2021). https://doi.org/10.1007/s13324-021-00501-8
Jahangiri, J.M., Ponnusamy, S.: Applications of subordination to functions with bounded boundary rotation. Arch. Math. 98, 173–182 (2012). https://doi.org/10.1007/s00013-012-0357-3
Kaplan, W.: Close-to-convex schlicht functions. Mich. Math. J. 1, 169–185 (1952). https://doi.org/10.1307/mmj/1028988895
Ruscheweyh, S.: Convolutions in Geometric Function Theory. Seminaire de Math. Sup. 83, Les Presses de laUniversité de Montreal (1982)
Sheil-Small, T.: The Hadamard product and linear transformations of classes of analytic functions. J. Anal. Math. 34, 204–239 (1978). https://doi.org/10.1007/BF02790013
Sheil-Small, T.: Some remarks on Bazilevič functions. J. Anal. Math. 43, 1–11 (1983). https://doi.org/10.1007/BF02790175
Sheil-Small, T.: Complex Polynomials. Cambridge University Press, New York(2002)
Funding
Not applicable
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no competing interests
Availability of data and material
Not applicable
Authors’ contributions
Not applicable
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ignaciuk, S., Parol, M. On the Koebe Quarter Theorem for certain polynomials of odd degree. Anal.Math.Phys. 12, 92 (2022). https://doi.org/10.1007/s13324-022-00703-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-022-00703-8