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1. 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 theoremCategories:53A10, 30D35, 53C42

2. 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 setCategory:30B40

3. 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 equationCategories:30D20, 32A15, 35F20

4. CMB Online first

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 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, MÃ¶bius invariant function spaceCategories:30H25, 31C25, 46E15

5. 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 TheoremCategories:55P15, 30G12

6. CMB Online first

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 matrixCategories:15A15, 30C10, 47A56

7. CMB 2016 (vol 59 pp. 776)

Gauthier, Paul M; Sharifi, Fatemeh
 The CarathÃ©odory Reflection Principle and Osgood-CarathÃ©odory Theorem on Riemann Surfaces The Osgood-CarathÃ©odory 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 CarathÃ©odory reflection principle. Keywords:bordered Riemann surface, reflection principle, Osgood-CarathÃ©odoryCategories:30C25, 30F99

8. 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 surfaceCategories:30E15, 30F99

9. 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, compactnessCategories:30H35, 28A12, 47B38, 46E15

10. 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 invariantCategories:20F67, 22E40, 30F40

11. 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 equivalenceCategories:32A38, 30H50, 32A10, 32M99

12. 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 theoremCategory:30E10

13. 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 theoremsCategories:30C80, 30C25, 30C99

14. 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 Lebesgue-points both for the weight function $w$ and for $\log w$. Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theoryCategories:42C05, 60B20, 30C85, 31A15

15. 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

16. 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 HilbertianCategories:53C23, 30L99

17. 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, uniquenessCategories:30B50, 11M41

18. 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 idealCategories:46J15, 46J20, 30H50

19. 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 domainCategories:32H02, 30C80, 32A30

20. 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 decompositionsCategories:30F10, 32G15, 53C22

21. 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 groupCategories:30F40, 22E40, 20H10

22. 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 radicalCategories:47L65, 47L20, 30J10, 30H50

23. 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 surfaceCategories:30F35, 30H25, 30H30

24. 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 sharingCategory:30D35

25. 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-functionCategories:30K05, 11M06
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