On the Koebe Quarter Theorem for certain polynomials

Abstract

AbstractWe study problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in $${\mathbb {T}}:=\{z\in {\mathbb {C}}:|z|=1\}$$ T : = { z ∈ C : | z | = 1 } . We found minimal disc containing all images of $${\mathbb {D}}:=\{z\in {\mathbb {C}}: |z|<1\}$$ D : = { z ∈ C : | z | < 1 } and maximal disc contained in all images of $${\mathbb {D}}$$ D through polynomials of degree 3 and 4. Moreover we determine the extremal functions for both problems.