Interior and exterior curves of finite Blaschke products

For a Blaschke product B of degree d and lambda on partial derivative D, let l(lambda) be the set of lines joining each distinct two preimages in B-1(A). The envelope of the family of lines {l(lambda)}(lambda is an element of partial derivative 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 L-lambda be the set of lines tangent to partial derivative D at the d preimages B-1(lambda) and the trace of the intersection points of each two elements in L-lambda as lambda 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. (C) 2018 Elsevier Inc. All rights reserved.