A Generalization of a Geometric Property of Blaschke Products to Selected Riemann Surfaces

Geometric properties of analytic functions provide deep insights into the structure of the domains on which they act. In particular, 2-degree Blaschke products exhibit a remarkable geometric feature in the unit disc: any two distinct points with equal images lie on a straight line passing through a fixed point. This striking collinearity property reveals a strong connection between algebraic expressions and geometric configurations. The present study aims to generalize this property beyond the unit disc, first to the upper half-plane and then to the compact Riemann surface associated with the multi-valued function √ z, thereby establishing a unified geometric framework.

The investigation proceeds in three stages. First, the classical geometric structure of 2-degree Blaschke products is analyzed using algebraic manipulation and geometric interpretation. Second, the unit disc is mapped conformally to the upper half-plane, where rational Nevanlinna functions are introduced as natural analogues. Their geometric behavior is studied to determine how their collinearity property transforms under this change of domain. Finally, the nonlinear mapping z = w ² and stereographic projection are employed to lift planar curves to the compactified Riemann surface of √ z, allowing a global geometric interpretation on the Riemann sphere.

It is shown that the collinearity property in the unit disc becomes a concyclicity property in the upper half-plane: points with equal function values lie on a circle passing through a fixed base point. Under the squaring map, chords of the unit disc transform into parabolas, which lift to spherical curves on the compact Riemann surface, all intersecting at infinity.

This study demonstrates that geometric properties of analytic functions are preserved and meaningfully transformed under conformal equivalence and compactification. The results highlight the role of Riemann surfaces in connecting algebraic structure, analytic behavior, and global geometry within a coherent framework.