We 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.
In this article we define a binary linear operator T for holomorphic functions in given open sets A and B in the complex plane under certain additional assumptions. It coincides with the classical Hadamard product of holomorphic functions in the case where A and B are the unit disk. We show that the operator T exists provided A and B are simply connected domains containing the origin. Moreover, T is determined explicitly by means of an integral form. To this aim we prove an alternative representation of the star product A * B of any sets A, B ⊂ C containing the origin. We also touch the problem of holomorphic extensibility of Hadamard product.
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