Interior and exterior curves of finite Blaschke products

For a Blaschke product B of degree d and λ on ∂D, let ℓλ be the set of lines joining each distinct two preimages in B−1(λ). The envelope of the family of lines {ℓλ}λ∈∂D is called the interior curve associated with B. In 2002, Daepp, Gorkin, and Mortini proved the interior curve associated with a Blaschke product of degree 3 forms an ellipse. While let Lλ be the set of lines tangent to ∂D at the d preimages B−1(λ) and the trace of the intersection points of each two elements in Lλ as λ ranges over the unit circle is called the exterior curve associated with B. In 2017, the author proved the exterior curve associated with a Blaschke product of degree 3 forms a non-degenerate conic. In this paper, for a Blaschke product of degree d, we give some geometrical properties that lie between the interior curve and the exterior curve.