TY - JOUR
T1 - An Extremal Problem for Odd Univalent Polynomials
AU - Dmitrishin, Dmitriy
AU - Gray, Daniel
AU - Stokolos, Alexander
AU - Tarasenko, Iryna
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.
PY - 2024/3
Y1 - 2024/3
N2 - For the univalent polynomials F(z)=∑j=1Najz2j-1 with real coefficients and normalization a1=1 we solve the extremal problem (Formula presented.) We show that the solution is (Formula presented.) and the extremal polynomial (Formula presented.) is unique and univalent, where Uj(x) is a Chebyshev polynomial of the second kind and Uj′(x) denotes the derivative. As an application, we obtain an estimate of the Koebe radius for odd univalent polynomials in D and formulate several conjectures.
AB - For the univalent polynomials F(z)=∑j=1Najz2j-1 with real coefficients and normalization a1=1 we solve the extremal problem (Formula presented.) We show that the solution is (Formula presented.) and the extremal polynomial (Formula presented.) is unique and univalent, where Uj(x) is a Chebyshev polynomial of the second kind and Uj′(x) denotes the derivative. As an application, we obtain an estimate of the Koebe radius for odd univalent polynomials in D and formulate several conjectures.
KW - Chebyshev polynomials
KW - Koebe one-quarter theorem
KW - Odd univalent polynomials
KW - T-folded Koebe function
UR - http://www.scopus.com/inward/record.url?scp=85163180757&partnerID=8YFLogxK
U2 - 10.1007/s40315-023-00487-3
DO - 10.1007/s40315-023-00487-3
M3 - Article
AN - SCOPUS:85163180757
SN - 1617-9447
VL - 24
SP - 83
EP - 100
JO - Computational Methods and Function Theory
JF - Computational Methods and Function Theory
IS - 1
ER -