In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.
In this paper, we first study the bounded mean oscillation of planar harmonic mappings, then a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings is established. At last, we obtain sharp estimates on Lipschitz
number of planar harmonic mappings in terms of bounded mean oscillation norm, which shows that the harmonic Bloch space is isomorphic to $BMO_{2}$ as a Banach space..
In this note, we consider the sufficient coefficient condition for some harmonic mappings in the unit disk which can be extended to the whole complex plane. As an application of this result, we will prove that a harmonic strongly starlike mapping has
a quasiconformal extension to the whole plane and will give an explicit form of its extension function. We also investigate the quasiconformal extension of harmonic mappings in the exterior unit disk.
We analyze a canonical construction of group-invariant CR Mappings between hyperquadrics due to DAngelo. Given source hyperquadric of $Q(1,1)$, we determine the signature of the target hyperquadric for all finite subgroups of $SU(1,1)$. We also exten
d combinatorial results proven by Loehr, Warrington, and Wilf on determinants of sparse circulant determinants. We apply these results to study CR mappings invariant under finite subgroups of $U(1,1)$.
In this paper we give some quantative characteristics of boundary asymptotic behavior of semigroups of holomorphic self-mappings of the unit disk including the limit curvature of their trajectories at the boundary Denjoy--Wolff point. This enable us
to establish an asymptotic rigidity property for semigroups of parabolic type.