Abstract: We study the Fourier series of circle homeomorphisms and circle embeddings, with an emphasis on the Blaschke product approximation and the vanishing of Fourier coefficients. The analytic properties of the Fourier series are related to the geometry of the circle embeddings, and have implications for the curvature of minimal surfaces. PubDate: 2019-03-11

Abstract: The present paper is devoted to finding the classes of complex periodic functions whose iterations map lines into dense subsets of the complex plane. We distinguish a class of functions such that its second iterations map almost all oblique lines into dense sets. A class of complex periodic functions are obtained whose third iterations of any oblique line generate dense subsets of the complex plane. PubDate: 2019-03-05

Abstract: In this paper, we study the existence of entire solutions of finite-order of non-linear difference equations of the form $$\begin{aligned} f^{n}(z)+q(z)\Delta _{c}f(z)=p_{1}\mathrm{e}^{\alpha _{1}z}+p_{2}\mathrm{e}^{\alpha _{2}z},\quad n\ge 2 \end{aligned}$$ and $$\begin{aligned} f^{n}(z)+q(z)\mathrm{e}^{Q(z)}f(z+c)=p_{1}\mathrm{e}^{\lambda z}+p_{2}\mathrm{e}^{-\lambda z},\quad n\ge 3 \end{aligned}$$ where q, Q are non-zero polynomials, \(c,\lambda ,p_{i},\alpha _{i}(i=1,2)\) are non-zero constants such that \(\alpha _{1}\ne \alpha _{2}\) and \(\Delta _{c}f(z)=f(z+c)-f(z)\not \equiv 0\) . Our results are improvements and complements of Wen et al. (Acta Math Sin 28:1295–1306, 2012), Yang and Laine (Proc Jpn Acad Ser A Math Sci 86:10–14, 2010) and Zinelâabidine (Mediterr J Math 14:1–16, 2017). PubDate: 2019-03-01

Abstract: It is shown that two key results on transcendental singularities for meromorphic functions of finite lower order have refinements which hold under the weaker hypothesis that the logarithmic derivative has finite lower order. PubDate: 2019-03-01

Abstract: In this paper, we study Fermat-type functional equations \(f^n+g^n+h^n=1\) in the complex plane. Alternative proofs of the known results for entire and meromorphic solutions of such equations are given. Moreover, some conditions on degrees of polynomial solutions are given. PubDate: 2019-03-01

Abstract: This paper presents a construction principle for the Schwarzian derivative of conformal mappings from an annulus onto doubly connected domains bounded by polygons of circular arcs. PubDate: 2019-03-01

Abstract: The aim of this work is to consider the bicomplex k-Pell quaternions and to present some properties involving this sequence, including the Binet-style formulae and the generating functions. Furthermore, Cassini’s identity, Catalan’s identity, and d’Ocagne’s identity for this type of bicomplex quaternions are given, and a different way to find the nth term of this sequence is stated using the determinant of a tridiagonal matrix whose entries are bicomplex k-Pell quaternions. PubDate: 2019-03-01

Abstract: For any finite Blaschke product B, there is an injective analytic map \(\varphi :{\mathbb {D}}\rightarrow {\mathbb {C}}\) and a polynomial p of the same degree as B such that \(B=p\circ \varphi \) on \({\mathbb {D}}\) . Several proofs of this result have been given over the past several years, using fundamentally different methods. However, even for low-degree Blaschke products, no method has hitherto been developed to explicitly compute the polynomial p or the associated conformal map \(\varphi \) . In this paper, we show how these functions may be computed for a Blaschke product of degree at most three, as well as for Blaschke products of arbitrary finite degree whose zeros are equally spaced on a circle centered at the origin. PubDate: 2019-03-01

Abstract: We study the growth of solutions of \(f''+A(z)f'+B(z)f=0\) , where A(z) and B(z) are non-trivial solutions of another second-order complex differential equations. Some conditions guaranteeing that every non-trivial solution of the equation is of infinite order are obtained, in which the notion of accumulation rays of the zero sequence of entire functions is used. PubDate: 2019-03-01

Abstract: We exploit the equality of Bergman analytic content and torsional rigidity of a simply connected domain to develop a new method for calculating these quantities. This method is particularly suitable for the case when the region in question is a polygon. A large number of examples are computed in explicit detail to demonstrate the utility of our ideas. PubDate: 2019-03-01

Abstract: An ancient optics problem of Ptolemy, studied later by Alhazen, is discussed. This problem deals with reflection of light in spherical mirrors. Mathematically, this reduces to the solution of a quartic equation, which we solve and analyze using a symbolic computation software. Similar problems have been recently studied in connection with ray-tracing, catadioptric optics, scattering of electromagnetic waves, and mathematical billiards, but we were led to this problem in our study of the so-called triangular ratio metric. PubDate: 2019-03-01

Abstract: Using algebraic methods, and motivated by the one variable case, we study a multipoint interpolation problem in the setting of several complex variables. The duality realized by the residue generator associated with an underlying Gorenstein algebra, using the Lagrange interpolation polynomial, plays a key role in the arguments. PubDate: 2019-03-01

Abstract: In this note, we compare a Gromov hyperbolic metric and the hyperbolic metric of the unit ball and obtain sharp inequalities between these two metrics, thus resolving a conjecture of Mohapatra and Sahoo. We also prove that the same inequalities hold for the above-mentioned metrics in the upper half space. PubDate: 2018-12-01

Abstract: It is well known that a domain in the plane is a quadrature domain with respect to area measure if and only if the function z extends meromorphically to the double, and it is a quadrature domain with respect to boundary arc length measure if and only if the complex unit tangent vector function T(z) extends meromorphically to the double. By applying the Cauchy integral formula to \(\bar{z}\) , we will shed light on the density of area quadrature domains among smooth domains with real analytic boundary. By extending \(\bar{z}\) and T(z) and applying the Cauchy integral formula to the Szegő kernel, we will obtain conformal mappings to nearby arc length quadrature domains and even domains that are like the unit disc in that they are simultaneously area and arc length quadrature domains. These “double quadrature domains” can be thought of as analogs of the unit disc in the multiply connected setting and the mappings so obtained as generalized Riemann mappings. The main theorems of this paper are not new, but the methods used in their proofs are new and more constructive than previous methods. The new computational methods give rise to numerical methods for computing generalized Riemann maps to nearby quadrature domains. PubDate: 2018-12-01

Abstract: Let \(\mathcal {A}_{0}\) denote the class of analytic functions f in the unit disk \({\mathbb {D}} = \{ z \in {\mathbb {C}} : z < 1\}\) normalized by \(f(0)=1\) . For \(f(z) = \sum \nolimits _{k=0}^{\infty } a_{k} z^{k}\) , the nth partial sum \(s_{n} (f,z)\) of f is defined by \(s_{n} (f,z) = \sum \nolimits _{k=0}^{n} a_{k} z^{k}\) , \(n=0,1,\ldots \) . A function \(f \in \mathcal {A}_{0}\) is said to be stable with respect to \(g \in \mathcal {A}_{0}\) if $$\begin{aligned} \frac{s_{n} (f,z)}{f(z)} \prec \frac{1}{g(z)}, \quad z \in {\mathbb {D}} \end{aligned}$$ holds for all \(n \in {\mathbb {N}}\) . In the present paper, we consider the following function $$\begin{aligned} v_{\lambda } (\alpha , z) := \left( \frac{1+(1-2\alpha ) z}{1-z}\right) ^{\lambda }, \end{aligned}$$ for \(0\le \alpha < 1\) and \(0\le \lambda \le 1\) . The aim of this paper is to prove that \(v_{\lambda }(\alpha ,z)\) is stable with respect to \(f_{\lambda } (z):= 1/(1-z)^{\lambda }\) for \(0<\lambda \le 1\) and \(1/2 \le \alpha < 1\) . Also, we prove that \(v_{\lambda }(\alpha ,z)\) is not stable with respect to itself when \(1/2< \alpha <1\) and \(0 <\lambda \le 1\) . Finally, we end this paper with two conjectures. PubDate: 2018-12-01

Authors:Trevor Richards Abstract: Let f be a meromorphic function with simply connected domain \(G\subset \mathbb {C}\) , and let \(\Gamma \subset \mathbb {C}\) be a smooth Jordan curve. We call a component of \(f^{-1}(\Gamma )\) in G a \(\Gamma \) -pseudo-lemniscate of f. In this note, we give criteria for a smooth Jordan curve \(\mathcal {S}\) in G (with bounded face D) to be a \(\Gamma \) -pseudo-lemniscate of f in terms of the number of preimages (counted with multiplicity) which a given w has under f in D (denoted \(\mathcal {N}_f(w)\) ), as w ranges over the Riemann sphere. As a corollary, we obtain the fact that if \(\mathcal {N}_f(w)\) takes three different value, then either \(\mathcal {S}\) contains a critical point of f, or \(f(\mathcal {S})\) is not a Jordan curve. PubDate: 2018-04-13 DOI: 10.1007/s40315-018-0242-6

Authors:Hervé Gaussier; Harish Seshadri Abstract: We prove that if a \({\mathcal {C}}^\infty \) -smooth bounded convex domain in \({\mathbb {C}}^n\) contains a holomorphic disc in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi distance. We also give examples of bounded smooth convex domains that are not strongly pseudoconvex but are Gromov hyperbolic. PubDate: 2018-04-13 DOI: 10.1007/s40315-018-0243-5

Authors:Md Firoz Ali; A. Vasudevarao Abstract: Let \((H,\oplus ,\odot )\) denote the Hornich space consisting of all locally univalent and analytic functions f on the unit disk \({\mathbb {D}}:=\{z\in {\mathbb {C}}:\, z <1\}\) with \(f(0)=0= f'(0)-1\) for which \(\arg f'\) is bounded in \({\mathbb {D}}\) . For \(f,g\in H\) and \(r,s\in \mathbb {R}\) , we consider the integral operator \(I_{r,s}(z):= \int _{0}^{z} (f'(\xi ))^r (g'(\xi ))^s\,\mathrm{d}\xi \) and determine all values of r and s for which the operator \((f,g)\mapsto I_{r,s}\) maps a specified subclass of H into another specified subclass of H. We also determine the set of extreme points for different subclasses of H with respect to the Hornich space structure. Using the extreme points, we develop a new approach to obtain the pre-Schwarzian norm estimate for different subclasses of H. We also consider a larger space \({\widetilde{H}}\) , whose linear structure is same as that of H and study the same problems as stated above for some subclasses of \({\widetilde{H}}\) . PubDate: 2018-03-27 DOI: 10.1007/s40315-018-0244-4

Authors:Xiaoguang Qi; Nan Li; Lianzhong Yang Abstract: This paper is devoted to studying some shared value properties for finite-order meromorphic solutions of the difference Painlevé IV equation. We also consider sharing value problems for the derivative of a meromorphic function f(z) with its shift \(f(z+c)\) and difference \(\Delta f\) . PubDate: 2018-03-26 DOI: 10.1007/s40315-018-0241-7