Angle duality and a gap principle for convex combinations of incomplete polynomials on the unit circle

In this paper, we establish an angle duality and a gap principle for convex combinations of incomplete polynomials, extending two results of Ge and Gonek in [IMRN, 2024].Our approach is geometric: we introduce an ``angle gain'' mechanism for points inside the lune region and quantify how moving away from the unit circle forces a definite increase in the relevant angle functional.This yields a robust lower bound that is uniform under convex mixing and leads to the desired separation phenomenon.The main difficulty is that incomplete polynomials and their convex combinations may have highly nonuniform root distributions on the unit circle, so classical convex-hull type constraints are too coarse; one must instead control the local geometry of chords and boundary arcs and relate it to critical-point behavior through sharp trigonometric inequalities.