It is shown that the Laurent series of a holomorphic function smooth up to the boundary on a Reinhardt domain in $mathbb{C}^n$ converges unconditionally to the function in the Fr{e}chet topology of the space of functions smooth up to the boundary.
Kalantaris Geometric Modulus Principle describes the local behavior of the modulus of a polynomial. Specifically, if $p(z) = a_0 + sum_{j=k}^n a_jleft(z-z_0right)^j,;a_0a_ka_n eq 0$, then the complex plane near $z = z_0$ comprises $2k$ sectors of angle $frac{pi}{k}$, alternating between arguments of ascent (angles $theta$ where $|p(z_0 + te^{itheta})| > |p(z_0)|$ for small $t$) and arguments of descent (where the opposite inequality holds). In this paper, we generalize the Geometric Modulus Principle to holomorphic and harmonic functions. As in Kalantaris original paper, we use these extensions to give succinct, elegant new proofs of some classical theorems from analysis.
We prove a theorem on separation of boundary null points for generators of continuous semigroups of holomorphic self-mappings of the unit disk in the complex plane. Our construction demonstrates the existence and importance of a particular role of the binary operation $circ$ given by $1 / f circ g = 1/f + 1/g$ on generators.
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.