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Search: MSC category 30 ( Functions of a complex variable )

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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 Gelfand-Naimark'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 Nyman-Beurling-Báez-Duarte criterion for the Riemann Hypothesis
A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty \left|1-\zeta A_N \left(\frac{1}{2}+it \right) \right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] 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 non-trivial zeros off the critical line.

Keywords:Riemann hypothesis, Riemann zeta function, Nyman-Beurling-Báez-Duarte 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 $$\Psi_B(z)=\sum_n \frac{1-|z_n|^2}{|1-\overline{z}_nz|^2}.$$ Moreover, it is a well-known 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 Carleson-Newman 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
Differential-free characterisation of smooth mappings with controlled growth
In this paper we give some generalizations and improvements of the Pavlović result on the Holland-Walsh 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, Holland-Walsh 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, operator-theoretic 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 $(1-|z|^2)^\alpha\,dA(z)$ 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

Ha, Pham Hoang; Kawakami, Yu
A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

Keywords:minimal surface, Gauss map, unicity theorem
Categories:53A10, 30D35, 53C42

8. CMB Online first

Gauthier, Paul M
The initial and terminal cluster sets of an analytic curve
For an analytic curve $\gamma:(a,b)\rightarrow \mathbb C,$ the set of values approached by $\gamma(t),$ as $t\searrow a$ and as $t\nearrow b$ can be any two continuua of $\mathbb C\cup\{\infty\}.$

Keywords:analytic curve, cluster set
Category:30B40

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 Möbius 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 non-Hilbert 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, Möbius invariant function space
Categories:30H25, 31C25, 46E15

11. CMB Online first

Hemasundar, Gollakota V. V.; Simha, R. R.
The Jordan Curve Theorem via Complex Analysis
The aim of this article is to give a proof of the Jordan Curve Theorem via complex analysis.

Keywords:Jordan Curve Theorem
Categories:55P15, 30G12

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)

Gauthier, Paul M; Sharifi, Fatemeh
The Carathéodory Reflection Principle and Osgood-Carathéodory Theorem on Riemann Surfaces
The Osgood-Carathéodory theorem asserts that conformal mappings between Jordan domains extend to homeomorphisms between their closures. For multiply-connected domains on Riemann surfaces, similar results can be reduced to the simply-connected case, but we find it simpler to deduce such results using a direct analogue of the Carathéodory reflection principle.

Keywords:bordered Riemann surface, reflection principle, Osgood-Carathéodory
Categories:30C25, 30F99

14. CMB 2016 (vol 60 pp. 300)

Gauthier, Paul M; Sharifi, Fatemeh
Luzin-type 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 one-parameter 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 non-negative 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 Szegő'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 Lebesgue-points 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
Non-branching 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 non-branching $RCD(0,N)$ geodesic spaces (and in particular, Alexandrov spaces) with full support measures. This is a special case of the Milnor's Conjecture for complete non-compact $RCD(0,N)$ spaces. One of the key tools we use is the Abresch-Gromoll type excess estimates for non-smooth spaces obtained by Gigli-Mosconi.

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)

Hu, Pei-Chu; Li, Bao Qin
A Simple Proof and Strengthening of a Uniqueness Theorem for L-functions
We give a simple proof and strengthening of a uniqueness theorem for functions in the extended Selberg class.

Keywords:meromorphic function, Dirichlet series, L-function, zero, order, uniqueness
Categories:30B50, 11M41

24. CMB 2015 (vol 58 pp. 350)

Merino-Cruz, 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 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
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