Let a < 0, Ω = ℂ -(-∞, a] and U = z: |z| < 1. We consider the class $S_H(U,\Omega )$ of functions f which are univalent, harmonic and sense preserving with f(U) = Ω and satisfy f(0) = 0, $f_z\left(0\right)>0$ and $f_{z̅}\left(0\right)=0$. We describe the closure $\overline{S_H(U,\Omega )}$ of $S_H(U,\Omega )$ and determine the extreme points of $\overline{S_H(U,\Omega )}$.

Let a < 0 < b and Ω(a,b) = ℂ - ((-∞, a] ∪ [b,+∞)) and U= z: |z| < 1. We consider the class $S_H(U,\Omega (a,b))$ of functions f which are univalent, harmonic and sense-preserving with f(U) = Ω and satisfying f(0) = 0, $f_z\left(0\right)>0$ and $f_z̅\left(0\right)=0$.

This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all...

This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all...

We establish certain properties for the class $𝒰(\Omega ,{\zeta }_0)$ of universal functions in $\Omega$ with respect to the center ${\zeta }_0\in \Omega$, for certain types of connected non-simply connected domains $\Omega$. In the case where $ℂ/\Omega$ is discrete we prove that this class is $G_{\delta }$-dense in $H\left(\Omega \right)$, depends on the center ${\zeta }_0$ and that the analog of Kahane’s conjecture does not hold.

Let $H\left(K\right)$ be the Hilbert space with reproducing kernel $K$. This paper characterizes some sufficient conditions for a sequence to be a universal interpolating sequence for $H\left(K\right)$.

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.

We strengthen a result of Chui and Parnes and we prove that the set of universal Taylor series is a $G_{\delta }$-dense subset of the space of holomorphic functions defined in the open unit disc. Our result provides the answer to a question stated by S.K. Pichorides concerning the limit set of Taylor series. Moreover, we study some properties of universal Taylor series and show, in particular, that they are trigonometric series in the sense of D. Menchoff.

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.

A generalized approach to several universality results is given by replacing holomorphic or harmonic functions by zero solutions of arbitrary linear partial differential operators. Instead of the approximation theorems of Runge and others, we use an approximation theorem of Hörmander.

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.