On the Koebe Quarter Theorem for certain polynomials

We study problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in T:={z∈C:|z|=1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}:=\{z\in {\mathbb {C}}:|z|=1\}$$\end{document}. We found minimal disc containing all images of D:={z∈C:|z|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}:=\{z\in {\mathbb {C}}: |z|<1\}$$\end{document} and maximal disc contained in all images of D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}$$\end{document} through polynomials of degree 3 and 4. Moreover we determine the extremal functions for both problems.