Let $C$ be the extended complex plane; $G\subset C$ a finite Jordan with $0\in G$; $w=\varphi \left(z\right)$ the conformal mapping of $G$ onto the disk $B\left(0;{\rho }_0\right):=\left\}w\left|w\right|<{\rho }_0\right\{$ normalized by $\varphi \left(0\right)=0$ and ${\varphi }^{\text{'}}\left(0\right)=1$. Let us set ${\varphi }_p\left(z\right):={\int }_0^z{\left[{\varphi }^{\text{'}}\left(\zeta \right)\right]}^{2/p}\mathrm{d}\zeta$, and let ${\pi }_{n,p}\left(z\right)$ be the generalized Bieberbach polynomial of degree $n$ for the pair $(G,0)$, which minimizes the integral $\underset{G}{\int \int }{\left|{\varphi }_p^{\text{'}}\left(z\right)-P_n^{\text{'}}\left(z\right)\right|}^p\mathrm{d}{\sigma }_z$ in the class of all polynomials of degree not exceeding $\le n$ with $P_n\left(0\right)=0$, $P_n^{\text{'}}\left(0\right)=1$. In this paper we study the uniform convergence of the generalized Bieberbach polynomials ${\pi }_{n,p}\left(z\right)$ to ${\varphi }_p\left(z\right)$ on $\overline{G}$ with interior and exterior zero angles and determine its dependence on the...

We prove the existence of sequences ${\varrho ₙ}_{n=1}^{\infty }$, ϱₙ → 0⁺, and ${zₙ}_{n=1}^{\infty }$, |zₙ| = 1/2, such that for every α ∈ ℝ and for every meromorphic function G(z) on ℂ, there exists a meromorphic function $F\left(z\right)=F_{G,\alpha }\left(z\right)$ on ℂ such that $\varrho {ₙ}^{\alpha }F(nzₙ+n\varrho ₙ\zeta )$ converges to G(ζ) uniformly on compact subsets of ℂ in the spherical metric. As a result, we construct a family of functions meromorphic on the unit disk that is $Q_m$-normal for no m ≥ 1 and on which an extension of Zalcman’s Lemma holds.

A holomorphic function $f$ on a simply connected domain $\Omega$ is said to possess a universal Taylor series about a point in $\Omega$ if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta $K$ outside $\Omega$ (provided only that $K$ has connected complement). This paper shows that this property is not conformally invariant, and, in the case where $\Omega$ is the unit disc, that such functions have extreme angular boundary behaviour.

We prove some conditions on a sequence of functions and on a complex domain for the existence of universal functions with respect to sequences of certain derivative and antiderivative operators related to them. Conditions for the equicontinuity of those families of operators are also studied. The conditions depend upon the "size" of the domain and functions. Some earlier results about multiplicative complex sequences are extended.