26. CMB 2015 (vol 58 pp. 381)
 Tang, Xiaomin; Liu, Taishun

The 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 selfmappings
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 

27. 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 SeppÃ¤lÃ¤. 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 

28. CMB 2012 (vol 56 pp. 881)
29. 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 

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

31. CMB 2012 (vol 56 pp. 769)
32. CMB 2012 (vol 56 pp. 544)
33. CMB 2012 (vol 56 pp. 241)
34. CMB 2011 (vol 56 pp. 229)
 Arvanitidis, Athanasios G.; Siskakis, Aristomenis G.

CesÃ ro Operators on the Hardy Spaces of the HalfPlane
In this article we study the CesÃ ro
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:CesÃ ro operators, Hardy spaces, semigroups, composition operators Categories:47B38, 30H10, 47D03 

35. CMB 2011 (vol 56 pp. 194)
 Stefánsson, Úlfar F.

On the Smallest and Largest Zeros of MÃ¼ntzLegendre Polynomials
MÃ¼ntzLegendre
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 $l_{n,n}\lt l_{n1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ 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 MÃ¼ntz space is
nondense 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:MÃ¼ntz polynomials, MÃ¼ntzLegendre polynomials Categories:42C05, 42C99, 41A60, 30B50 

36. 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 nonextendable. 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 

37. 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
$(1z'^2)^{\beta\alpha}u(z')\ge\delta$, where $d(\cdot,\cdot)$ denotes
the pseudodistance on $D$.
Keywords:$\alpha$Bloch function, multiplication operator Categories:32A18, 30H05 

38. 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
\[
\prod_{n=1}^{\infty} \frac{(1q^n)^5}{(1q^{5n})}
=15\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n
\]
and
\[
q\prod_{n=1}^{\infty} \frac{(1q^{5n})^5}{(1q^{n})}
=\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n.
\]
We give several more identities of this type.
Keywords:Power series expansions of certain infinite products Categories:11E25, 11F11, 11F27, 30B10 

39. CMB 2009 (vol 52 pp. 53)
 Cummins, C. J.

Cusp Forms Like $\Delta$
Let $f$ be a squarefree 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 

40. 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 ``nonEuclidean 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 

41. CMB 2008 (vol 51 pp. 497)
 Borwein, Peter; Choi, KwokKwong Stephen; Mercer, Idris

Expected Norms of ZeroOne Polynomials
Let $\cA_n = \big\{ a_0 + a_1 z + \cdots + a_{n1}z^{n1} : a_j \in \{0, 1 \
} \big\}$, whose elements are called \emf{zeroone polynomials}
and correspond naturally to the $2^n$ subsets of $[n] := \{ 0, 1,
\ldots, n1 \}$. 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 zeroone 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 $c_k
:= \sum_{j=0}^{Nk} a_j a_{j+k}$ 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 $1
\le k \le n1$ 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 

42. CMB 2008 (vol 51 pp. 334)
43. CMB 2008 (vol 51 pp. 195)
44. 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 

45. CMB 2007 (vol 50 pp. 123)
46. CMB 2007 (vol 50 pp. 11)
 Borwein, David; Borwein, Jonathan

van der Pol Expansions of LSeries
We provide concise series representations for various
Lseries integrals. Different techniques are needed below and above
the abscissa of absolute convergence of the underlying Lseries.
Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, Lseries Categories:11M35, 11M41, 30B50 

47. CMB 2006 (vol 49 pp. 438)
 Mercer, Idris David

Unimodular Roots of\\ Special Littlewood Polynomials
We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n1} z^{n1}$ a Littlewood
polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ selfreciprocal
if $\alpha(z) = z^{n1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if
$n = 2m+1$ and $a_{m+j} = (1)^j a_{mj}$ 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 selfreciprocal
Littlewood polynomial must have a zero on the unit circle.
Categories:26C10, 30C15, 42A05 

48. CMB 2006 (vol 49 pp. 381)
49. 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 

50. CMB 2005 (vol 48 pp. 409)