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26. CMB 2004 (vol 47 pp. 17)

Gorkin, Pamela; Mortini, Raymond
Universal Singular Inner Functions
We show that there exists a singular inner function $S$ which is universal for noneuclidean translates; that is one for which the set $\{S(\frac{z+z_n}{1+\bar z_nz}):n\in\mathbb{N}\}$ is locally uniformly dense in the set of all zero-free holomorphic functions in $\mathbb{D}$ bounded by one.

Category:30D50

27. CMB 2003 (vol 46 pp. 559)

Marco, Nicolas; Massaneda, Xavier
On Density Conditions for Interpolation in the Ball
In this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of $\C^n$, $n>1$. We first give density conditions for a sequence to be interpolating for the class $A^{-\infty}$ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension $1$, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for $n=1$ coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space.

Categories:32A36, 32A38, 30E05

28. CMB 2003 (vol 46 pp. 95)

Gauthier, P. M.
Cercles de remplissage for the Riemann Zeta Function
The celebrated theorem of Picard asserts that each non-constant entire function assumes every value infinitely often, with at most one exception. The Riemann zeta function has this Picard behaviour in a sequence of discs lying in the critical band and whose diameters tend to zero. According to the Riemann hypothesis, the value zero would be this (unique) exceptional value.

Keywords:cercles de remplissage, Riemann zeta function
Category:30

29. CMB 2002 (vol 45 pp. 265)

Nawrocki, Marek
On the Smirnov Class Defined by the Maximal Function
H.~O.~Kim has shown that contrary to the case of $H^p$-space, the Smirnov class $M$ defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, {\it i.e.} they have the same dual spaces and the same Fr\'echet envelopes. We describe a general form of a continuous linear functional on $M$ and multiplier from $M$ into $H^p$, $0 < p \leq \infty$.

Keywords:Smirnov class, maximal radial function, multipliers, dual space, Fréchet envelope
Categories:46E10, 30A78, 30A76

30. CMB 2002 (vol 45 pp. 89)

Grant, David
On Gunning's Prime Form in Genus $2$
Using a classical generalization of Jacobi's derivative formula, we give an explicit expression for Gunning's prime form in genus 2 in terms of theta functions and their derivatives.

Categories:14K25, 30F10

31. CMB 2002 (vol 45 pp. 154)

Weitsman, Allen
On the Poisson Integral of Step Functions and Minimal Surfaces
Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

Keywords:harmonic mappings, dilatation, minimal surfaces
Categories:30C62, 31A05, 31A20, 49Q05

32. CMB 2002 (vol 45 pp. 36)

Cummins, C. J.
Modular Equations and Discrete, Genus-Zero Subgroups of $\SL(2,\mathbb{R})$ Containing $\Gamma(N)$
Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains $\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there is a unique normalised generator of the field of $G$-automorphic functions which is known as a normalised Hauptmodul. This paper gives a characterisation of normalised Hauptmoduls as formal $q$ series using modular polynomials.

Categories:11F03, 11F22, 30F35

33. CMB 2001 (vol 44 pp. 420)

Gauthier, P. M.; Pouryayevali, M. R.
Approximation by Meromorphic Functions With Mittag-Leffler Type Constraints
Functions defined on closed sets are simultaneously approximated and interpolated by meromorphic functions with prescribed poles and zeros outside the set of approximation.

Categories:30D30, 30E10, 30E15

34. CMB 2000 (vol 43 pp. 183)

Ionesei, Gheorghe
A Gauge Theoretic Proof of the Abel-Jacobi Theorem
We present a new, simple proof of the classical Abel-Jacobi theorem using some elementary gauge theoretic arguments.

Keywords:Abel-Jacobi theorem, abelian gauge theory
Categories:58D27, 30F99

35. CMB 2000 (vol 43 pp. 115)

Schmutz Schaller, Paul
Perfect Non-Extremal Riemann Surfaces
An infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group $\PSL(2,{\sf Z})$ of level $6$. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function {\it syst\/} which associates to every Riemann surface $M$ the length of a systole, which is a shortest closed geodesic of $M$.

Categories:11H99, 11F06, 30F45

36. CMB 2000 (vol 43 pp. 105)

Overholt, Marius
Sets of Uniqueness for Univalent Functions
We observe that any set of uniqueness for the Dirichlet space $\cD$ is a set of uniqueness for the class $S$ of normalized univalent holomorphic functions.

Categories:30C55, 30C15

37. CMB 1999 (vol 42 pp. 139)

Bonet, José; Domański, Paweł; Lindström, Mikael
Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions
Every weakly compact composition operator between weighted Banach spaces $H_v^{\infty}$ of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space $H_v^{\infty}$ are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

Keywords:weighted Banach spaces of holomorphic functions, composition operator, compact operator, weakly compact operator
Categories:47B38, 30D55, 46E15

38. CMB 1999 (vol 42 pp. 3)

Beauzamy, Bernard
How the Roots of a Polynomial Vary with Its Coefficients: A Local Quantitative Result
A well-known result, due to Ostrowski, states that if $\Vert P-Q \Vert_2< \varepsilon$, then the roots $(x_j)$ of $P$ and $(y_j)$ of $Q$ satisfy $|x_j -y_j|\le C n \varepsilon^{1/n}$, where $n$ is the degree of $P$ and $Q$. Though there are cases where this estimate is sharp, it can still be made more precise in general, in two ways: first by using Bombieri's norm instead of the classical $l_1$ or $l_2$ norms, and second by taking into account the multiplicity of each root. For instance, if $x$ is a simple root of $P$, we show that $|x-y|< C \varepsilon$ instead of $\varepsilon^{1/n}$. The proof uses the properties of Bombieri's scalar product and Walsh Contraction Principle.

Category:30C10

39. CMB 1998 (vol 41 pp. 473)

Müller, Jürgen; Wengenroth, Jochen
Separating singularities of holomorphic functions
We present a short proof for a classical result on separating singularities of holomorphic functions. The proof is based on the open mapping theorem and the fusion lemma of Roth, which is a basic tool in complex approximation theory. The same method yields similar separation results for other classes of functions.

Categories:30E99, 30E10

40. CMB 1997 (vol 40 pp. 475)

Lou, Zengjian
Coefficient multipliers of Bergman spaces $A^p$, II
We show that the multiplier space $(A^1,X)=\{g:M_\infty(r,g'') =O(1-r)^{-1}\}$, where $X$ is $\BMOA$, $\VMOA$, $B$, $B_0$ or disk algebra $A$. We give the multipliers from $A^1$ to $A^q(H^q)(1\le q\le \infty)$, we also give the multipliers from $l^p(1\le p\le 2), C_0, \BMOA$, and $H^p(2\le p<\infty)$ into $A^q(1\le q\le 2)$.

Keywords:Multiplier, Bergman space, Hardy space, Bloch space, $\BMOA$.
Categories:30H05, 30B10

41. CMB 1997 (vol 40 pp. 356)

Mazet, Pierre
Principe du maximum et lemme de Schwarz, a valeurs vectorielles
Nous {\'e}tablissons un th{\'e}or{\`e}me pour les fonctions holomorphes {\`a} valeurs dans une partie convexe ferm{\'e}e. Ce th{\'e}or{\`e}me pr{\'e}cise la position des coefficients de Taylor de telles fonctions et peut {\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce th{\'e}or{\`e}me permet de retrouver des versions connues du principe du maximum et d'obtenir de nouveaux r{\'e}sultats sur les applications holomorphes {\`a} valeurs vectorielles.

Keywords:Principe du maximum, lemme de Schwarz, points extr{émaux.
Categories:30C80, 32A30, 46G20, 52A07

42. CMB 1997 (vol 40 pp. 271)

Bergweiler, Walter
Non-real periodic points of entire functions
It is shown that if $f$ is an entire transcendental function, $l$ a straight line in the complex plane, and $n\geq 2$, then $f$ has infinitely many repelling periodic points of period $n$ that do not lie on $l$.

Categories:30D05, 58F23
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