1. CMB Online first
 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 

2. CMB Online first
 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 

3. CMB Online first
 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 

4. CMB Online first
 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 

5. CMB Online first
 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 

6. CMB Online first
 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 Online first
8. CMB Online first
9. CMB Online first
 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 

10. 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 

11. CMB Online first
12. 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 

13. CMB 2016 (vol 59 pp. 776)
14. 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 

15. 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 

16. 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 

17. 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 

18. 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 

19. 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 

20. 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 

21. CMB 2015 (vol 59 pp. 13)
 Aulaskari, Rauno; Chen, Huaihui

On classes $Q_p^\#$ for Hyperbolic Riemann surfaces
The $Q_p$ spaces of holomorphic functions on
the disk, hyperbolic Riemann surfaces or complex unit ball have
been studied deeply.
Meanwhile, there are a lot of papers devoted to the $Q^\#_p$
classes of meromorphic functions on the disk or hyperbolic Riemann
surfaces. In this paper, we prove the nesting property (inclusion
relations) of $Q^\#_p$ classes on hyperbolic Riemann surfaces.
The same property for $Q_p$ spaces was also established systematically
and precisely in
earlier work
by the authors of this paper.
Keywords:$Q_p^\#$ class, hyperbolic Riemann surface, spherical Dirichlet function, Categories:30D50, 30F35 

22. CMB 2015 (vol 58 pp. 787)
 Kitabeppu, Yu; Lakzian, Sajjad

Nonbranching RCD$(0,N)$ Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups
In this paper, we generalize the finite generation result of
Sormani
to nonbranching $RCD(0,N)$
geodesic spaces (and in particular, Alexandrov spaces) with full
support measures. This is a special case of the Milnor's Conjecture
for complete noncompact $RCD(0,N)$ spaces. One of the key tools
we use is the AbreschGromoll type excess estimates for nonsmooth
spaces obtained by GigliMosconi.
Keywords:Milnor conjecture, non negative Ricci curvature, curvature dimension condition, finitely generated, fundamental group, infinitesimally Hilbertian Categories:53C23, 30L99 

23. CMB 2015 (vol 59 pp. 119)
24. CMB 2015 (vol 58 pp. 350)
 MerinoCruz, Héctor; Wawrzynczyk, Antoni

On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions
We recently introduced a weighted Banach algebra $\mathfrak{A}_G^n$ of
functions which are holomorphic on the unit disc $\mathbb{D}$, continuous
up to the boundary and of the class $C^{(n)}$ at all points where
the function $G$ does not vanish. Here, $G$ refers to a function
of the disc algebra without zeros on $\mathbb{D}$. Then we proved that
all closed ideals in $\mathfrak{A}_G^n$ with at most countable hull are
standard. In the present paper, on the assumption that $G$ is
an outer function in $C^{(n)}(\overline{\mathbb{D}})$ having infinite roots
in $\mathfrak{A}_G^n$ and countable zero set $h(G)$, we show that all the
closed ideals $I$ with hull containing $h(G)$ are standard.
Keywords:Banach algebra, disc algebra, holomorphic spaces, standard ideal Categories:46J15, 46J20, 30H50 

25. 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 
