1. CMB Online first
 Elmadani, Y.; Labghail, I.

Cyclicity in Dirichlet spaces
Let $\mu$ be a positive finite Borel measure on the unit circle
and $\mathcal{D}(\mu)$ the associated harmonically weighted Dirichlet
space. In this paper we show that for each closed subset $E$
of the unit circle with zero $c_{\mu}$capacity, there exists
a function $f\in\mathcal{D}(\mu)$ such that $f$ is cyclic (i.e.,
$\{p f: p $ is a polynomial$\}$ is dense in $\mathcal{D}(\mu)$),
$f$ vanishes on $E$, and $f$ is uniformly continuous.
Then we provide a sufficient
condition for a continuous function on the closed unit disk to
be cyclic in $\mathcal{D}(\mu)$.
Keywords:Dirichlettype space, cyclic vector, capacity, strongtype inequality Categories:47B38, 30C85, 30H05 

2. CMB Online first
 Kalvin, Victor; Kokotov, Alexey

Determinant of the Laplacian on tori of constant positive curvature with one conical point
We find an explicit expression for the zetaregularized determinant
of (the Friedrichs extensions) of the Laplacians on a compact
Riemann surface of genus one with conformal metric of curvature
$1$ having a single conical singularity of angle $4\pi$.
Keywords:determinant of Laplacian, moduli space, spectral zetafunction, curvature one, conical point, conical singularity, Riemann surface, compact Riemann surface Categories:47A10, 58J52, 30F45 

3. CMB Online first
 Alam, Ihab Al; Lefèvre, Pascal

Embeddings of MÃ¼ntz Spaces in $L^\infty(\mu)$
In this paper, we discuss the properties of the embedding
operator $i^\Lambda_\mu : M_\Lambda^\infty\hookrightarrow L^\infty(\mu),$
where $\mu$ is a positive Borel measure on $[0,1]$ and $M_{\Lambda}^{\infty}$
is a MÃ¼ntz space. In particular, we compute the essential norm
of this embedding. As a consequence, we recover some results
of
the first author.
We also study the compactness (resp. weak compactness)
and compute the essential norm (resp. generalized essential norm)
of the embedding $i_{\mu_1,\,\mu_2} : L^\infty(\mu_1)\hookrightarrow
L^\infty(\mu_2)$, where $\mu_1$, $\mu_2$ are two positive Borel
measures on $[0,1]$ with $\mu_2$ absolutely continuous with respect
to $\mu_1$.
Keywords:MÃ¼ntz space, embedding, essential norm, compact operator Categories:32C22, 47B33, 30B10 

4. CMB Online first
 Wang, Jianfei; Zhang, Danli

Extension operators for biholomorphic mappings
Suppose that $D\subset\mathbb{C}$ is a simply connected subdomain
containing the origin and $f(z_1)$ is a normalized convex (resp.,
starlike) function on $D$. Let
$$
\Omega_{N}(D)=\{(z_1,w_1,\ldots,w_k)\in \mathbb{C}\times{\mathbb{C}}^{n_1}\times\cdots\times{\mathbb{C}}^{n_k}:
\w_1\_{p_1}^{p_1}+\cdots+\w_k\_{p_k}^{p_k}<\frac{1}{\lambda_{D}(z_1)}\},$$
where $p_j\geq 1$, $N=1+n_1+\cdots+n_k,\,w_1\in{\mathbb{C}}^{n_1},\ldots,w_k\in{\mathbb{C}}^{n_k}$
and $\lambda_{D}$ is the density of the hyperbolic metric on
$D$. In this paper, we prove that
\begin{equation*}
\Phi_{N,{1/p_{1}},\cdots,{1}/{p_{k}}}(f)(z_1,w_1,\ldots,w_k)=\big(f(z_{1}),
(f'(z_{1}))^{1/p_{1}}w_1,\cdots,(f'(z_{1}))^{1/p_{k}}w_k\big)
\end{equation*}
is a normalized convex (resp., starlike) mapping on $\Omega_{N}(D)$.
If $D$ is the unit disk, then our result reduces to Gong and
Liu \cite{GL2} via a new method. Moreover, we give a new operator
for
convex mapping construction on an unbounded domain in ${\mathbb{C}}^{2}$.
By using a geometric approach, we prove that $\Phi_{N,{1/p_{1}},\cdots,{1}/{p_{k}}}(f)$
is a spirallike mapping of type $\alpha$ when $f$ is
a spirallike function of type $\alpha$ on the unit disk.
Keywords:biholomorphic mapping, $\varepsilon$starlike mapping, spirallike mapping Categories:32H02, 30C45 

5. CMB 2018 (vol 61 pp. 738)
 CruzUribe, David; Rodney, Scott; Rosta, Emily

PoincarÃ© Inequalities and Neumann Problems for the $p$Laplacian
We prove an equivalence between weighted PoincarÃ© inequalities
and
the existence of weak solutions to a Neumann problem related
to a
degenerate $p$Laplacian. The PoincarÃ© inequalities are
formulated in the context of degenerate Sobolev spaces defined
in
terms of a quadratic form, and the associated matrix is the
source of
the degeneracy in the $p$Laplacian.
Keywords:degenerate Sobolev space, $p$Laplacian, PoincarÃ© inequalities Categories:30C65, 35B65, 35J70, 42B35, 42B37, 46E35 

6. CMB 2018 (vol 61 pp. 509)
 Cui, Xiaohui; Wang, Chunjie; Zhu, Kehe

Area Integral Means of Analytic Functions in the Unit Disk
For an analytic function $f$ on the unit disk $\mathbb D$ we show that
the $L^2$ integral mean of $f$ on $c\lt z\lt r$ with
respect to the weighted area measure $(1z^2)^\alpha\,dA(z)$
is a logarithmically convex function of $r$ on $(c,1)$,
where $3\le\alpha\le0$ and $c\in[0,1)$. Moreover, the range
$[3,0]$ for $\alpha$ is best possible. When
$c=0$, our arguments here also simplify the proof for several
results we obtained in earlier papers.
Keywords:logarithmic convexity, area integral mean, Bergman space, Hardy space Categories:30H10, 30H20 

7. CMB 2018 (vol 61 pp. 622)
 Maier, Helmut; Rassias, Michael Th.

On the size of an expression in the NymanBeurlingBÃ¡ezDuarte criterion for the Riemann Hypothesis
A crucial role in the NymanBeurlingBÃ¡ezDuarte approach to
the Riemann Hypothesis is played by the distance
\[
d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{\infty}^\infty
\left1\zeta A_N
\left(\frac{1}{2}+it
\right)
\right^2\frac{dt}{\frac{1}{4}+t^2}\:,
\]
where the infimum is over all Dirichlet polynomials
$$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$
of length $N$.
In this paper we investigate $d_N^2$ under the assumption that
the Riemann zeta function has four nontrivial zeros off the
critical line.
Keywords:Riemann hypothesis, Riemann zeta function, NymanBeurlingBÃ¡ezDuarte criterion Categories:30C15, 11M26 

8. CMB 2017 (vol 61 pp. 704)
 Bénéteau, Catherine Anne; Fleeman, Matthew C.; Khavinson, Dmitry S.; Seco, Daniel; Sola, Alan

Remarks on Inner Functions and Optimal Approximants
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.
Keywords:inner function, reproducing Kernel Hilbert Space, operatortheoretic function theory Categories:46E22, 30J05 

9. CMB 2017 (vol 61 pp. 70)
 Dang, Pei; Liu, Hua; Qian, Tao

Hilbert Transformation and Representation of the $ax+b$ Group
In this paper we study the Hilbert transformations over
$L^2(\mathbb{R})$
and $L^2(\mathbb{T})$ from
the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$
commutative with the ax+b group we show that the operator is
of the form
$
\lambda I+\eta H,
$
where $I$ and $H$ are the identity operator and Hilbert transformation
respectively, and $\lambda,\eta$ are complex numbers. In the
related literature this result was proved through first invoking
the boundedness result of the operator, proved though a big
machinery.
In our setting the boundedness is a consequence of the boundedness
of the Hilbert transformation. The methodology that we use is
GelfandNaimark's representation of the ax+b group. Furthermore
we prove a similar result on the unit circle. Although there
does not exist a group like ax+b on the unit circle, we construct
a semigroup to play the same symmetry role for the Hilbert transformations
over the circle $L^2(\mathbb{T}).$
Keywords:singular integral, Hilbert transform, the $ax+b$ group Categories:30E25, 44A15, 42A50 

10. CMB 2017 (vol 61 pp. 640)
 Reijonen, Atte

Remark on integral means of derivatives of Blaschke products
If $B$ is the Blachke product with zeros $\{z_n\}$, then $B'(z)\le
\Psi_B(z)$, where
$$\Psi_B(z)=\sum_n \frac{1z_n^2}{1\overline{z}_nz^2}.$$
Moreover, it is a wellknown fact that, for $0\lt p\lt \infty$,
$$M_p(r,B')=
\left(\frac{1}{2\pi}\int_{0}^{2\pi} B'(re^{i\t})^p\,d\t
\right)^{1/p}, \quad 0\le r\lt 1,$$
is bounded if and only if $M_p(r,\Psi_B)$ is bounded.
We find a Blaschke product $B_0$ such that $M_p(r,B_0')$ and
$M_p(r,\Psi_{B_0})$ are not comparable for any $\frac12\lt p\lt \infty$.
In addition, it is shown that, if $0\lt p\lt \infty$, $B$ is a CarlesonNewman
Blaschke product and a weight $\omega$ satisfies a certain regularity
condition, then
$$
\int_\mathbb{D} B'(z)^p\omega(z)\,dA(z)\asymp \int_\mathbb{D} \Psi_B(z)^p\omega(z)\,dA(z),
$$
where $dA(z)$ is the Lebesgue area measure on the unit disc.
Keywords:Bergman space, Blaschke product, Hardy space, integral mean Categories:30J10, 30H10, 30H20 

11. CMB 2017 (vol 61 pp. 628)
 Marković, Marijan

Differentialfree characterisation of smooth mappings with controlled growth
In this paper we give some generalizations and
improvements of the PavloviÄ result on the
HollandWalsh type characterization of the Bloch space of
continuously differentiable (smooth) functions in
the unit ball in $\mathbf{R}^m$.
Keywords:Bloch type space, Lipschitz type space, HollandWalsh characterisation, hyperbolic distance, analytic function, Mobius transform Categories:32A18, 30D45 

12. CMB 2017 (vol 61 pp. 292)
13. CMB 2017 (vol 61 pp. 282)
14. CMB 2017 (vol 61 pp. 142)
 Li, Bao Qin

An Equivalent Form of Picard's Theorem and Beyond
This paper gives an equivalent form of Picard's
theorem via entire solutions of the functional equation $f^2+g^2=1$,
and then its improvements and applications to certain nonlinear
(ordinary and partial) differential equations.
Keywords:entire function, Picard's Theorem, functional equation, partial differential equation Categories:30D20, 32A15, 35F20 

15. CMB 2017 (vol 60 pp. 690)
 Bao, Guanlong; Göğüş, Nıhat Gökhan; Pouliasis, Stamatis

$\mathcal{Q}_p$ Spaces and Dirichlet Type Spaces
In this paper, we show that the MÃ¶bius invariant
function space $\mathcal {Q}_p$ can be generated by variant
Dirichlet type spaces
$\mathcal{D}_{\mu, p}$ induced by finite positive Borel measures
$\mu$ on the open unit disk. A criterion for the equality between
the space $\mathcal{D}_{\mu, p}$ and the usual Dirichlet type
space $\mathcal {D}_p$ is given. We obtain a sufficient condition
to construct different $\mathcal{D}_{\mu, p}$ spaces
and we provide examples.
We establish decomposition theorems for $\mathcal{D}_{\mu,
p}$ spaces, and prove that the nonHilbert space $\mathcal
{Q}_p$ is equal to the intersection of Hilbert spaces $\mathcal{D}_{\mu,
p}$. As an application of the relation between $\mathcal {Q}_p$
and $\mathcal{D}_{\mu, p}$ spaces, we also obtain that there
exist different $\mathcal{D}_{\mu, p}$ spaces; this is a trick
to prove the existence without constructing examples.
Keywords:$\mathcal {Q}_p$ space, Dirichlet type space, MÃ¶bius invariant function space Categories:30H25, 31C25, 46E15 

16. CMB 2017 (vol 61 pp. 124)
17. CMB 2017 (vol 60 pp. 561)
 Kurdyka, Krzysztof; Paunescu, Laurentiu

Nuij Type Pencils of Hyperbolic Polynomials
Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic
(i.e. has only real roots) then $p+sp'$ is also hyperbolic for
any
$s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials
of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$.
We give a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic
polynomials.
We give also a full characterization of those $a= (a_1, \dots,
a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$
admit universal determinantal representations. In fact we show
that all these sequences come from special symmetric Toeplitz
matrices.
Keywords:hyperbolic polynomial, stable polynomial, determinantal representa tion, symmetric Toeplitz matrix Categories:15A15, 30C10, 47A56 

18. CMB 2016 (vol 59 pp. 776)
19. CMB 2016 (vol 60 pp. 300)
 Gauthier, Paul M; Sharifi, Fatemeh

Luzintype Holomorphic Approximation on Closed Subsets of Open Riemann Surfaces
It is known that if $E$ is a closed subset of an open Riemann
surface $R$ and $f$ is a holomorphic function on a neighbourhood
of $E,$ then it is ``usually" not possible to approximate $f$
uniformly by functions holomorphic on all of $R.$ We show, however,
that for every open Riemann surface $R$ and every closed subset
$E\subset R,$ there is closed subset $F\subset E,$ which approximates
$E$ extremely well, such that every function holomorphic on $F$
can be approximated much better than uniformly by functions holomorphic
on $R$.
Keywords:Carleman approximation, tangential approximation, Myrberg surface Categories:30E15, 30F99 

20. CMB 2016 (vol 59 pp. 878)
 Wang, Jianfei

The Carleson Measure Problem Between Analytic Morrey Spaces
The purpose of this paper is to characterize positive measure
$\mu$ on the unit disk such that the analytic
Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly
embedded to the tent space
$\mathcal{T}_{q,1\frac{q}{p}(1\eta)}^{\infty}(\mu)$ for the
case $1\leq q\leq p\lt \infty$
respectively. As an application, these results are used to
establish the boundedness and compactness of integral operators
and multipliers between analytic Morrey spaces.
Keywords:Morrey space, Carleson measure problem, boundedness, compactness Categories:30H35, 28A12, 47B38, 46E15 

21. CMB 2016 (vol 59 pp. 244)
 Cao, Wensheng; Huang, Xiaolin

A Note on Quaternionic Hyperbolic Ideal Triangle Groups
In this paper, the quaternionic hyperbolic
ideal triangle groups are parameterized by a real oneparameter
family $\{\phi_s: s\in \mathbb{R}\}$. The indexing parameter $s$ is
the tangent of the quaternionic angular invariant of a triple
of points in $\partial \mathbf{H}_{\mathbb{h}}^2 $ forming this ideal
triangle. We show that if $s \gt \sqrt{125/3}$ then $\phi_s$ is
not a discrete embedding, and if $s \leq \sqrt{35}$
then $\phi_s$ is a discrete embedding.
Keywords:quaternionic inversion, ideal triangle group, quaternionic Cartan angular invariant Categories:20F67, 22E40, 30F40 

22. CMB 2016 (vol 59 pp. 346)
 Krantz, Steven

On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains
We study and generalize a classical theorem of L. Bers that classifies
domains up to biholomorphic equivalence in terms of the algebras
of
holomorphic functions on those domains. Then we develop applications
of these results to the study of domains with noncompact automorphism
group.
Keywords:Bers's theorem, algebras of holomorphic functions, noncompact automorphism group, biholomorphic equivalence Categories:32A38, 30H50, 32A10, 32M99 

23. CMB 2015 (vol 59 pp. 87)
 Gauthier, Paul M.; Kienzle, Julie

Approximation of a Function and its Derivatives by Entire Functions
A simple proof is given for the fact that, for $m$ a nonnegative
integer, a function $f\in C^{(m)}(\mathbb{R}),$ and an arbitrary positive
continuous function $\epsilon,$ there is an entire function $g,$
such that $g^{(i)}(x)f^{(i)}(x)\lt \epsilon(x),$ for all $x\in\mathbb{R}$
and for each $i=0,1\dots,m.$ We also consider the situation,
where $\mathbb{R}$ is replaced by an open interval.
Keywords:Carleman theorem Category:30E10 

24. CMB 2015 (vol 59 pp. 30)
 Cleanthous, Galatia

A Geometric Extension of Schwarz's Lemma and Applications
Let $f$ be a holomorphic function of the unit
disc $\mathbb{D},$ preserving the origin. According to Schwarz's
Lemma, $f'(0)\leq1,$ provided that $f(\mathbb{D})\subset\mathbb{D}.$
We prove that this bound still holds, assuming only that $f(\mathbb{D})$
does not contain any closed rectilinear segment
$[0,e^{i\phi}],\;\phi\in[0,2\pi],$ i.e. does not contain any
entire radius of the closed unit disc. Furthermore, we apply
this result to the hyperbolic density and we give a covering
theorem.
Keywords:Schwarz's Lemma, polarization, hyperbolic density, covering theorems Categories:30C80, 30C25, 30C99 

25. CMB 2015 (vol 59 pp. 211)
 Totik, Vilmos

Universality Under SzegÅ's Condition
This paper presents a
theorem on universality on orthogonal polynomials/random matrices
under a weak local condition on the weight function $w$.
With a new inequality for
polynomials and with the use of fast decreasing polynomials,
it is shown that an approach of
D. S. Lubinsky is applicable. The proof works
at all points which are Lebesguepoints both
for the weight function $w$ and for $\log w$.
Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theory Categories:42C05, 60B20, 30C85, 31A15 
