  This paper mainly focuses on the CR analogue of the three-circle theorem in a
complete noncompact pseudohermitian manifold of vanishing torsion being odd
dimensional counterpart of K\"ahler geometry. In this paper, we show that the
CR three-circle theorem holds if its pseudohermitian sectional curvature is
nonnegative. As an application, we confirm the first CR Yau's uniformization
conjecture and obtain the CR analogue of the sharp dimension estimate for CR
holomorphic functions of polynomial growth and its rigidity when the
pseudohermitian sectional curvature is nonnegative. This is also the first step
toward second and third CR Yau's uniformization conjecture. Moreover, in the
course of the proof of the CR three-circle theorem, we derive CR sub-Laplacian
comparison theorem. Then Liouville theorem holds for positive pseudoharmonic
functions in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing
torsion and nonnegative pseudohermitian Ricci curvature.
