### A gap series with growth conditions and its applications.

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Let $\mathcal{A}$ denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F\left(0\right)=0,{F}^{\text{'}}\left(0\right)=1$, whereas $A\subset \mathcal{A}$ is an arbitrarily fixed subset. In this paper various properties of the classes ${A}_{\alpha},\alpha \in C\{-1,-\frac{1}{2},...\}$, of functions of the form $f=F*{k}_{\alpha}$ are studied, where $F\in .A$, ${k}_{\alpha}\left(z\right)=k(z,\alpha )=z+\frac{1}{1+\alpha}{z}^{2}+...+\frac{1}{1+(n-1)\alpha}{z}^{n}+...$, and $F*{k}_{\alpha}$ denotes the Hadamard product of the functions $F$ and ${k}_{\alpha}$. Some special cases of the set $A$ were considered by other authors (see, for example, [15],[6],[3]).

We construct a nonbasic harmonic mapping of the unit disk onto a convex wedge. This mapping satisfies the partial differential equation ${f}_{\overline{z}}=a{f}_{z}$ where a(z) is a nontrivial extreme point of the unit ball of ${H}^{\infty}$.

We prove that each degree two quasiregular polynomial is conjugate to Q(z) = z² - (p+q)|z|² + pqz̅² + c, |p| < 1, |q| < 1. We also show that the complexification of Q can be extended to a polynomial endomorphism of ℂℙ² which acts as a Blaschke product (z-p)/(1-p̅z) · (z-q)/(1-q̅z) on ℂℙ²∖ℂ². Using this fact we study the dynamics of Q under iteration.

We give a simple algebraic condition on the leading homogeneous term of a polynomial mapping from ℝ² into ℝ² which is equivalent to the fact that the complexification of this mapping can be extended to a polynomial endomorphism of ℂℙ². We also prove that this extension acts on ℂℙ²∖ℂ² as a quotient of finite Blaschke products.