TY - JOUR
T1 - An extremal problem for polynomials
AU - Dmitrishin, Dmitriy
AU - Smorodin, Andrey
AU - Stokolos, Alex
N1 - Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2022/1
Y1 - 2022/1
N2 - For the polynomials F(z)=∑j=1Najzj with real coefficients and normalization a1=1 we solve the extremal problem supa2,…,aN(infz∈D{Re(F(z)):Im(F(z))=0}). We show that the solution is [Formula presented], and the extremal polynomial [Formula presented] is unique and univalent, where the Uj(x) are the Chebyshev polynomials of the second kind, j=1,…,N. As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.
AB - For the polynomials F(z)=∑j=1Najzj with real coefficients and normalization a1=1 we solve the extremal problem supa2,…,aN(infz∈D{Re(F(z)):Im(F(z))=0}). We show that the solution is [Formula presented], and the extremal polynomial [Formula presented] is unique and univalent, where the Uj(x) are the Chebyshev polynomials of the second kind, j=1,…,N. As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.
KW - Chebyshev polynomials
KW - Extremal problems
KW - Fejér-Riesz representation
KW - Koebe one-quarter theorem
KW - Variational methods
UR - http://www.scopus.com/inward/record.url?scp=85120084795&partnerID=8YFLogxK
U2 - 10.1016/j.acha.2021.08.008
DO - 10.1016/j.acha.2021.08.008
M3 - Article
AN - SCOPUS:85120084795
SN - 1063-5203
VL - 56
SP - 283
EP - 305
JO - Applied and Computational Harmonic Analysis
JF - Applied and Computational Harmonic Analysis
ER -