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1. CMB Online first

Tang, Xiaomin; Liu, Taishun
Schwarz Lemma at the Boundary of the Egg Domain $B_{p_1, p_2}$ in $\mathbb{C}^n$
Let $B_{p_1, p_2}=\{z\in\mathbb{C}^n: |z_1|^{p_1}+|z_2|^{p_2}+\cdots+|z_n|^{p_2}\lt 1\}$ be an egg domain in $\mathbb{C}^n$. In this paper, we first characterize the Kobayashi metric on $B_{p_1, p_2}\,(p_1\geq 1, p_2\geq 1)$, and then establish a new type of the classical boundary Schwarz lemma at $z_0\in\partial{B_{p_1, p_2}}$ for holomorphic self-mappings of $B_{p_1, p_2}(p_1\geq 1, p_2\gt 1)$, where $z_0=(e^{i\theta}, 0, \dots, 0)'$ and $\theta\in \mathbb{R}$.

Keywords:holomorphic mapping, Schwarz lemma, Kobayashi metric, egg domain
Categories:32H02, 30C80, 32A30

2. CMB 2013 (vol 57 pp. 870)

Parlier, Hugo
A Short Note on Short Pants
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.

Keywords:hyperbolic surfaces, geodesics, pants decompositions
Categories:30F10, 32G15, 53C22

3. CMB 2012 (vol 56 pp. 881)

Xie, BaoHua; Wang, JieYan; Jiang, YuePing
Free Groups Generated by Two Heisenberg Translations
In this paper, we will discuss the groups generated by two Heisenberg translations of $\mathbf{PU}(2,1)$ and determine when they are free.

Keywords:free group, Heisenberg group, complex triangle group
Categories:30F40, 22E40, 20H10

4. CMB 2012 (vol 57 pp. 80)

Khemphet, Anchalee; Peters, Justin R.
Semicrossed Products of the Disk Algebra and the Jacobson Radical
We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.

Keywords:semicrossed product, disk algebra, Jacobson radical
Categories:47L65, 47L20, 30J10, 30H50

5. CMB 2012 (vol 56 pp. 466)

Aulaskari, Rauno; Rättyä, Jouni
Inclusion Relations for New Function Spaces on Riemann Surfaces
We introduce and study some new function spaces on Riemann surfaces. For certain parameter values these spaces coincide with the classical Dirichlet space, BMOA or the recently defined $Q_p$ space. We establish inclusion relations that generalize earlier known inclusions between the above-mentioned spaces.

Keywords:Bloch space, BMOA, $Q_p$, Green's function, hyperbolic Riemann surface
Categories:30F35, 30H25, 30H30

6. CMB 2012 (vol 56 pp. 769)

Lahiri, Indrajit; Kaish, Imrul
A Non-zero Value Shared by an Entire Function and its Linear Differential Polynomials
In this paper we study uniqueness of entire functions sharing a non-zero finite value with linear differential polynomials and address a result of W. Wang and P. Li.

Keywords:entire function, linear differential polynomial, value sharing
Category:30D35

7. CMB 2012 (vol 56 pp. 241)

Betsakos, Dimitrios; Pouliasis, Stamatis
Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius
We prove variants of Schwarz's lemma involving monotonicity properties of condenser capacity and inner radius. Also, we examine when a similar monotonicity property holds for the hyperbolic metric.

Keywords:condenser capacity, inner radius, hyperbolic metric, Schwarz's lemma
Categories:30C80, 30F45, 31A15

8. CMB 2012 (vol 56 pp. 544)

Gauthier, P. M.
Universally Overconvergent Power Series via the Riemann Zeta-function
The Riemann zeta-function is employed to generate universally overconvergent power series.

Keywords:overconvergence, zeta-function
Categories:30K05, 11M06

9. CMB 2011 (vol 56 pp. 229)

Arvanitidis, Athanasios G.; Siskakis, Aristomenis G.
Cesàro Operators on the Hardy Spaces of the Half-Plane
In this article we study the Cesàro operator $$ \mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta, $$ and its companion operator $\mathcal{T}$ on Hardy spaces of the upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$ as resolvents for appropriate semigroups of composition operators and we find the norm and the spectrum in each case. The relation of $\mathcal{C}$ and $\mathcal{T}$ with the corresponding Ces\`{a}ro operators on Lebesgue spaces $L^p(\mathbb R)$ of the boundary line is also discussed.

Keywords:Cesàro operators, Hardy spaces, semigroups, composition operators
Categories:47B38, 30H10, 47D03

10. CMB 2011 (vol 56 pp. 194)

Stefánsson, Úlfar F.
On the Smallest and Largest Zeros of Müntz-Legendre Polynomials
Müntz-Legendre polynomials $L_n(\Lambda;x)$ associated with a sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in $L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s are distinct, it is well known that $L_n(\Lambda;x)$ has exactly $n$ zeros $l_{n,n}\lt l_{n-1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ on $(0,1)$. First we prove the following global bound for the smallest zero, $$ \exp\biggl(-4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}. $$ An important consequence is that if the associated Müntz space is non-dense in $L_2[0,1]$, then $$ \inf_{n}x_{n,n}\geq \exp\biggl({-4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0, $$ so the elements $L_n(\Lambda;x)$ have no zeros close to 0. Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed, $$ \lim_{n\rightarrow\infty} \vert \log l_{k,n}\vert \sum_{j=0}^n (2\lambda_j+1)= \Bigl(\frac{j_k}{2}\Bigr)^2, $$ where $j_k$ denotes the $k$-th zero of the Bessel function $J_0$.

Keywords:Müntz polynomials, Müntz-Legendre polynomials
Categories:42C05, 42C99, 41A60, 30B50

11. CMB 2011 (vol 55 pp. 509)

Gauthier, P. M.; Nestoridis, V.
Domains of Injective Holomorphy
A domain $\Omega$ is called a domain of injective holomorphy if there exists an injective holomorphic function $f\colon \Omega\rightarrow\mathbb{C}$ that is non-extendable. We give examples of domains that are domains of injective holomorphy and others that are not. In particular, every regular domain $(\overline\Omega^o=\Omega)$ is a domain of injective holomorphy, and every simply connected domain is a domain of injective holomorphy as well.

Keywords:domains of holomorphy
Category:30Exx

12. CMB 2009 (vol 53 pp. 23)

Chen, Huaihui; Zhang, Minzhu
Boundedness From Below of Multiplication Operators Between $\alpha$-Bloch Spaces
In this paper, the boundedness from below of multiplication operators between $\alpha$-Bloch spaces $\mathcal B^\alpha$, $\alpha\gt 0$, on the unit disk $D$ is studied completely. For a bounded multiplication operator $M_u\colon \mathcal B^\alpha\to\mathcal B^\beta$, defined by $M_uf=uf$ for $f\in\mathcal B^\alpha$, we prove the following result: (i) If $0\lt \beta\lt \alpha$, or $0\lt \alpha\le1$ and $\alpha\lt \beta$, $M_u$ is not bounded below; (ii) if $0\lt \alpha=\beta\le1$, $M_u$ is bounded below if and only if $\liminf_{z\to\partial D}|u(z)|\gt 0$; (iii) if $1\lt \alpha\le\beta$, $M_u$ is bounded below if and only if there exist a $\delta\gt 0$ and a positive $r\lt 1$ such that for every point $z\in D$ there is a point $z'\in D$ with the property $d(z',z)\lt r$ and $(1-|z'|^2)^{\beta-\alpha}|u(z')|\ge\delta$, where $d(\cdot,\cdot)$ denotes the pseudo-distance on $D$.

Keywords:$\alpha$-Bloch function, multiplication operator
Categories:32A18, 30H05

13. CMB 2009 (vol 52 pp. 481)

Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.
Some Infinite Products of Ramanujan Type
In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities \[ \prod_{n=1}^{\infty} \frac{(1-q^n)^5}{(1-q^{5n})} =1-5\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n \] and \[ q\prod_{n=1}^{\infty} \frac{(1-q^{5n})^5}{(1-q^{n})} =\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n. \] We give several more identities of this type.

Keywords:Power series expansions of certain infinite products
Categories:11E25, 11F11, 11F27, 30B10

14. CMB 2009 (vol 52 pp. 53)

Cummins, C. J.
Cusp Forms Like $\Delta$
Let $f$ be a square-free integer and denote by $\Gamma_0(f)^+$ the normalizer of $\Gamma_0(f)$ in $\SL(2,\R)$. We find the analogues of the cusp form $\Delta$ for the groups $\Gamma_0(f)^+$.

Categories:11F03, 11F22, 30F35

15. CMB 2008 (vol 51 pp. 497)

Borwein, Peter; Choi, Kwok-Kwong Stephen; Mercer, Idris
Expected Norms of Zero-One Polynomials
Let $\cA_n = \big\{ a_0 + a_1 z + \cdots + a_{n-1}z^{n-1} : a_j \in \{0, 1 \ } \big\}$, whose elements are called \emf{zero-one polynomials} and correspond naturally to the $2^n$ subsets of $[n] := \{ 0, 1, \ldots, n-1 \}$. We also let $\cA_{n,m} = \{ \alf(z) \in \cA_n : \alf(1) = m \}$, whose elements correspond to the ${n \choose m}$ subsets of~$[n]$ of size~$m$, and let $\cB_n = \cA_{n+1} \setminus \cA_n$, whose elements are the zero-one polynomials of degree exactly~$n$. Many researchers have studied norms of polynomials with restricted coefficients. Using $\norm{\alf}_p$ to denote the usual $L_p$ norm of~$\alf$ on the unit circle, one easily sees that $\alf(z) = a_0 + a_1 z + \cdots + a_N z^N \in \bR[z]$ satisfies $\norm{\alf}_2^2 = c_0$ and $\norm{\alf}_4^4 = c_0^2 + 2(c_1^2 + \cdots + c_N^2)$, where $c_k := \sum_{j=0}^{N-k} a_j a_{j+k}$ for $0 \le k \le N$. If $\alf(z) \in \cA_{n,m}$, say $\alf(z) = z^{\beta_1} + \cdots + z^{\beta_m}$ where $\beta_1 < \cdots < \beta_m$, then $c_k$ is the number of times $k$ appears as a difference $\beta_i - \beta_j$. The condition that $\alf \in \cA_{n,m}$ satisfies $c_k \in \{0,1\}$ for $1 \le k \le n-1$ is thus equivalent to the condition that $\{ \beta_1, \ldots, \beta_m \}$ is a \emf{Sidon set} (meaning all differences of pairs of elements are distinct). In this paper, we find the average of~$\|\alf\|_4^4$ over $\alf \in \cA_n$, $\alf \in \cB_n$, and $\alf \in \cA_{n,m}$. We further show that our expression for the average of~$\|\alf\|_4^4$ over~$\cA_{n,m}$ yields a new proof of the known result: if $m = o(n^{1/4})$ and $B(n,m)$ denotes the number of Sidon sets of size~$m$ in~$[n]$, then almost all subsets of~$[n]$ of size~$m$ are Sidon, in the sense that $\lim_{n \to \infty} B(n,m)/\binom{n}{m} = 1$.

Categories:11B83, 11C08, 30C10

16. CMB 2008 (vol 51 pp. 481)

Bayart, Frédéric
Universal Inner Functions on the Ball
It is shown that given any sequence of automorphisms $(\phi_k)_k$ of the unit ball $\bn$ of $\cn$ such that $\|\phi_k(0)\|$ tends to $1$, there exists an inner function $I$ such that the family of ``non-Euclidean translates" $(I\circ\phi_k)_k$ is locally uniformly dense in the unit ball of $H^\infty(\bn)$.

Keywords:inner functions, automorphisms of the ball, universality
Categories:32A35, 30D50, 47B38

17. CMB 2008 (vol 51 pp. 334)

Ascah-Coallier, I.; Gauthier, P. M.
Value Distribution of the Riemann Zeta Function
In this note, we give a new short proof of the fact, recently discovered by Ye, that all (finite) values are equidistributed by the Riemann zeta function.

Keywords:Nevanlinna theory, deficiency, Riemann zeta function
Category:30D35

18. CMB 2008 (vol 51 pp. 195)

Chen, Huaihui; Gauthier, Paul
Boundedness from Below of Composition Operators on $\alpha$-Bloch Spaces
We give a necessary and sufficient condition for a composition operator on an $\alpha$-Bloch space with $\alpha\ge 1$ to be bounded below. This extends a known result for the Bloch space due to P. Ghatage, J. Yan, D. Zheng, and H. Chen.

Keywords:Bloch functions, composition operators
Categories:32A18, 30H05

19. CMB 2007 (vol 50 pp. 579)

Kot, Piotr
$p$-Radial Exceptional Sets and Conformal Mappings
For $p>0$ and for a given set $E$ of type $G_{\delta}$ in the boundary of the unit disc $\partial\mathbb D$ we construct a holomorphic function $f\in\mathbb O(\mathbb D)$ such that \[ \int_{\mathbb D\setminus[0,1]E}|ft|^{p}\,d\mathfrak{L}^{2}<\infty\] and\[ E=E^{p}(f)=\Bigl\{ z\in\partial\mathbb D:\int_{0}^{1}|f(tz)|^{p}\,dt=\infty\Bigr\} .\] In particular if a set $E$ has a measure equal to zero, then a function $f$ is constructed as integrable with power $p$ on the unit disc $\mathbb D$.

Keywords:boundary behaviour of holomorphic functions, exceptional sets
Categories:30B30, 30E25

20. CMB 2007 (vol 50 pp. 11)

Borwein, David; Borwein, Jonathan
van der Pol Expansions of L-Series
We provide concise series representations for various L-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying L-series.

Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, L-series
Categories:11M35, 11M41, 30B50

21. CMB 2007 (vol 50 pp. 123)

Nikolov, Nikolai; Pflug, Peter
Simultaneous Approximation and Interpolation on Arakelian Sets
We extend results of P.~M. Gauthier, W. Hengartner and A.~A. Nersesyan on simultaneous approximation and interpolation on Arakelian sets.

Keywords:Arakelian's theorem,, Arakelian sets
Category:30E10

22. CMB 2006 (vol 49 pp. 438)

Mercer, Idris David
Unimodular Roots of\\ Special Littlewood Polynomials
We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if $n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ for all $j$. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle.

Categories:26C10, 30C15, 42A05

23. CMB 2006 (vol 49 pp. 381)

Girela, Daniel; Peláez, José Ángel
On the Membership in Bergman Spaces of the Derivative of a Blaschke Product With Zeros in a Stolz Domain
It is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces $A^p$ with $01$). As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that $B'\notin A^{3/2}$.

Keywords:Blaschke products, Hardy spaces, Bergman spaces
Categories:30D50, 30D55, 32A36

24. CMB 2005 (vol 48 pp. 580)

Kot, Piotr
Exceptional Sets in Hartogs Domains
Assume that $\Omega$ is a Hartogs domain in $\mathbb{C}^{1+n}$, defined as $\Omega=\{(z,w)\in\mathbb{C}^{1+n}:|z|<\mu(w),w\in H\}$, where $H$ is an open set in $\mathbb{C}^{n}$ and $\mu$ is a continuous function with positive values in $H$ such that $-\ln\mu$ is a strongly plurisubharmonic function in $H$. Let $\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$. For a given set $E$ contained in $H$ of the type $G_{\delta}$ we construct a holomorphic function $f\in\mathbb{O}(\Omega)$ such that \[ E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}|f(\cdot\,,w)|^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}. \]

Keywords:boundary behaviour of holomorphic functions,, exceptional sets
Category:30B30

25. CMB 2005 (vol 48 pp. 409)

Gauthier, P. M.; Xiao, J.
The Existence of Universal Inner Functions on the Unit Ball of $\mathbb{C}^n$
It is shown that there exists an inner function $I$ defined on the unit ball ${\bf B}^n$ of ${\mathbb C}^n$ such that each function holomorphic on ${\bf B}^n$ and bounded by $1$ can be approximated by ``non-Euclidean translates" of $I$.

Keywords:universal inner functions
Categories:32A35, 30D50, 47B38
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