TY - JOUR

T1 - Univalent polynomials and Koebe’s one-quarter theorem

AU - Dmitrishin, Dmitriy

AU - Dyakonov, Konstantin

AU - Stokolos, Alex

N1 - Publisher Copyright:
© 2019, Springer Nature Switzerland AG.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - The famous Koebe 14 theorem deals with univalent (i.e., injective) analytic functions f on the unit disk D. It states that if f is normalized so that f(0) = 0 and f′(0) = 1 , then the image f(D) contains the disk of radius 14 about the origin, the value 14 being best possible. Now suppose f is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises? And for which polynomials is it attained? A plausible conjecture is stated, and the case of small degrees is settled.

AB - The famous Koebe 14 theorem deals with univalent (i.e., injective) analytic functions f on the unit disk D. It states that if f is normalized so that f(0) = 0 and f′(0) = 1 , then the image f(D) contains the disk of radius 14 about the origin, the value 14 being best possible. Now suppose f is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises? And for which polynomials is it attained? A plausible conjecture is stated, and the case of small degrees is settled.

KW - Koebe radius

KW - Koebe’s one-quarter theorem

KW - Univalent polynomial

UR - http://www.scopus.com/inward/record.url?scp=85065148978&partnerID=8YFLogxK

U2 - 10.1007/s13324-019-00305-x

DO - 10.1007/s13324-019-00305-x

M3 - Article

AN - SCOPUS:85065148978

SN - 1664-2368

VL - 9

SP - 991

EP - 1004

JO - Analysis and Mathematical Physics

JF - Analysis and Mathematical Physics

IS - 3

ER -