Authors:David Minda Pages: 579 - 590 Abstract: In this paper, precise versions of several intuitive properties of quotients of hyperbolic metrics are established. Suppose that \(\Omega _j\) is a hyperbolic region in \(\mathbb {C}_\infty = \mathbb {C}\cup \{\infty \}\) with hyperbolic metric \(\lambda _j\) , \(j=1,2\) , and \(\Omega _1 \subsetneq \Omega _2\) . First, it is shown that \(\lambda _1/ \lambda _2 \approx 1\) on compact subsets of \(\Omega _1\) that are not too close to \(\partial \Omega _1\) . Second, \(\lambda _1/ \lambda _2 \approx 1\) when z is near \((\partial \Omega _1 \;\cap \; \partial \Omega _2 ) {\setminus } F_b\) , where \(F = \partial \Omega _1 \;\cap \;\Omega _2\) and \(F_b = {{\mathrm{cl}}}(F)\;\cap \;\Omega _2\) . The main tools used in establishing these results are sharp elementary bounds for \(\lambda _1(z)/ \lambda _2(z)\) in terms of the hyperbolic distance relative to \(\Omega _2\) from z to \(\partial \Omega _1 \;\cap \;\Omega _2\) that were first established and employed in complex dynamics. PubDate: 2017-12-01 DOI: 10.1007/s40315-017-0195-1 Issue No:Vol. 17, No. 4 (2017)

Authors:Andreas Schweizer Pages: 591 - 601 Abstract: Let \(\mathcal{{F}}\) be a family of meromorphic functions on a domain D. We present a quite general sufficient condition for \(\mathcal{{F}}\) to be a normal family. This criterion contains many known results as special cases. The overall idea is that certain comparatively weak conditions on \(\mathcal{{F}}\) by local arguments lead to somewhat stronger conditions, which in turn lead to even stronger conditions on the limit function g in the famous Zalcman Lemma. Ultimately, the defect relations for g force normality of \(\mathcal{{F}}\) . PubDate: 2017-12-01 DOI: 10.1007/s40315-017-0196-0 Issue No:Vol. 17, No. 4 (2017)

Authors:Juha-Matti Huusko; María J. Martín Pages: 603 - 612 Abstract: In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that \(\limsup _{ z \rightarrow 1} S(f)(z) (1- z ^2)^2 < 2\) , then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that \(\limsup _{ z \rightarrow 1} f''(z)/f'(z) \, (1- z ^2)< 1\) . In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. PubDate: 2017-12-01 DOI: 10.1007/s40315-017-0197-z Issue No:Vol. 17, No. 4 (2017)

Authors:Thi Hoai An Ta; Viet Phuong Nguyen Pages: 613 - 634 Abstract: Consider meromorphic functions f, g, and \(\alpha ,\) where \(\alpha \) is a small function with respect to f and g. Let Q be a polynomial of one variable. We give suitable conditions on the degree of Q and on the number of zeros and the multiplicities of the zeros of \(Q'\) so as to be able to conclude uniqueness results if differential polynomials of the form \((Q(f))^{(k)}\) and \((Q(g))^{(k)}\) share \(\alpha \) counting multiplicities. We do not assume that Q has a large order zero, nor do we place restrictions on the zeros and poles of \(\alpha .\) Thus, our work improves on many prior results that either assume Q has a high order zero or place restrictions on the small function \(\alpha \) . PubDate: 2017-12-01 DOI: 10.1007/s40315-017-0198-y Issue No:Vol. 17, No. 4 (2017)

Authors:Masayo Fujimura Pages: 635 - 652 Abstract: We study geometrical properties of finite Blaschke products. For a Blaschke product B of degree d, let \(L_{\lambda }\) be the set of the lines tangent to the unit circle at the d preimages \( B^{-1}(\lambda ) \) . We show that the trace of the intersection points of each pair of two elements in \( L_{\lambda } \) as \( \lambda \) ranges over the unit circle forms an algebraic curve of degree at most \( d-1 \) . In case of low degree, we have more precise results. For instance, for \( d=3 \) , the trace forms a conic section. For \( d=4 \) , we provide a necessary and sufficient condition for Blaschke products whose trace include a conic section. PubDate: 2017-12-01 DOI: 10.1007/s40315-017-0201-7 Issue No:Vol. 17, No. 4 (2017)

Authors:Hu Chunying; Shi Qingtian Pages: 653 - 662 Abstract: A new kind of functional, analogous to the Douglas–Dirichlet functional, is defined as $$\begin{aligned} E'[f]=\displaystyle \iint _{\Omega }\sigma (z)( f_{z} ^{2}+ f_{\overline{z}} ^{2})\mathrm{d}x\mathrm{d}y \end{aligned}$$ for \(f\in {C^{2}}\) on \(\Omega \) with a conformal metric density \(\sigma (z)\) . A critical point of this new functional is said to be a \(\sigma (z)\) -harmonic mapping. We consider the harmonicity of the inverse function of a \(\sigma (z)\) -harmonic diffeomorphism and obtain a necessary and sufficient condition, which improves on the corresponding result for Euclidean harmonic mappings. In addition, a property of the inverse function of \(\rho \) -harmonic mappings is investigated and an example is given. PubDate: 2017-12-01 DOI: 10.1007/s40315-017-0202-6 Issue No:Vol. 17, No. 4 (2017)

Authors:Yuk-J. Leung Pages: 663 - 678 Abstract: We continue our investigation on a second variation formula of the Koebe function in the class \(\Sigma \) of functions analytic and univalent in the exterior of the unit disk. Our aim is to give some supporting evidence of a conjecture raised by William Kirwan on the coefficients of functions in this class. PubDate: 2017-12-01 DOI: 10.1007/s40315-017-0204-4 Issue No:Vol. 17, No. 4 (2017)

Authors:Ilgiz R Kayumov; Saminathan Ponnusamy Pages: 679 - 688 Abstract: We determine the Bohr radius for the class of odd functions f satisfying \( f(z) \le 1\) for all \( z <1\) , solving the recent problem of Ali et al. (J Math Anal Appl 449(1):154–167, 2017). In fact, we solve this problem in a more general setting. Then we discuss Bohr’s radius for the class of analytic functions g, when g is subordinate to a member of the class of odd univalent functions. PubDate: 2017-12-01 DOI: 10.1007/s40315-017-0206-2 Issue No:Vol. 17, No. 4 (2017)

Authors:Jerry R. Muir Pages: 715 - 733 Abstract: An integral formula of Cauchy type was recently developed that reproduces any continuous \(f:\overline{{\mathbb {B}}} \rightarrow {\mathbb {C}}^n\) that is holomorphic in the open unit ball \({\mathbb {B}}\) of \({\mathbb {C}}^n\) using a fixed vector-valued kernel and the scalar expression \(\langle f(u),u \rangle \) , where \(u\in \partial {\mathbb {B}}\) and \(\langle \cdot ,\cdot \rangle \) is the Hermitian inner product in \({\mathbb {C}}^n\) , which is key to defining the numerical range of f. We consider Hardy-type spaces associated with this vector-valued kernel. In particular, we introduce spaces of vector-valued holomorphic mappings properly containing the vector-valued Hardy spaces that are reproduced through the process described above and isomorphic spaces of scalar-valued non-holomorphic functions that satisfy many of the familiar properties of Hardy space functions. In the spirit of providing a straightforward introduction to these spaces, proof techniques have been kept as elementary as possible. In particular, the theory of maximal functions and singular integrals is avoided. PubDate: 2017-12-01 DOI: 10.1007/s40315-017-0203-5 Issue No:Vol. 17, No. 4 (2017)

Authors:Alan R. Legg Abstract: We make use of the Bergman kernel function to study quadrature domains whose quadrature identities hold for \(L^2\) holomorphic functions of several complex variables. We generalize some mapping properties of planar quadrature domains and point out some differences from the planar case. We then show that every smooth bounded convex domain in \({\mathbb {C}}^n\) is biholomorphic to a quadrature domain. Finally, the possibility of continuous deformations within the class of planar quadrature domains is examined. PubDate: 2017-11-22 DOI: 10.1007/s40315-017-0224-0

Authors:Iason Efraimidis Abstract: A conjecture of Bombieri (Invent Math 4:26–67, 1967) states that the coefficients of a normalized univalent function f should satisfy $$\begin{aligned} \liminf _{f\rightarrow K} \frac{n-\mathrm{Re\,}a_n}{m-\mathrm{Re\,}a_m} = \min _{t\in {\mathbb {R}}} \, \frac{n\sin t -\sin (nt)}{m\sin t -\sin (mt)}, \end{aligned}$$ when f approaches the Koebe function \(K(z)=\frac{z}{(1-z)^2}\) . Recently, Leung [10] disproved this conjecture for \(n=2\) and for all \(m\ge 3\) and, also, for \(n=3\) and for all odd \(m\ge 5\) . Complementing his work, we disprove it for all \(m>n\ge 2\) which are simultaneously odd or even and, also, for the case when m is odd, n is even and \(n\le \frac{m+1}{2}\) . We mostly not only make use of trigonometry but also employ Dieudonné’s criterion for the univalence of polynomials. PubDate: 2017-11-07 DOI: 10.1007/s40315-017-0222-2

Authors:Ran-Ran Zhang; Zhi-Bo Huang Abstract: In this paper, using the theory of linear algebra, we investigate the non-linear difference equation of the following form in the complex plane: $$\begin{aligned} f(z)^n + p(z)f(z+\eta ) = \beta _1e^{\alpha _1z}+\beta _2e^{\alpha _2z}+\cdots +\beta _se^{\alpha _sz}, \end{aligned}$$ where n, s are the positive integers, \(p(z)\not \equiv 0\) is a polynomial and \(\eta , \beta _1, \ldots , \beta _s, \alpha _1, \ldots , \alpha _s\) are the constants with \(\beta _1 \ldots \beta _s\alpha _1 \ldots \alpha _s\ne 0\) , and show that this equation just has meromorphic solutions with hyper-order at least one when \(n\ge 2+s\) . Other cases are also obtained. PubDate: 2017-11-07 DOI: 10.1007/s40315-017-0223-1

Authors:Igor E. Pritsker Abstract: We study the asymptotic distribution of zeros for the random rational functions that can be viewed as partial sums of a random Laurent series. If this series defines a random analytic function in an annulus A, then the zeros accumulate on the boundary circles of A, being equidistributed in the angular sense, with probability 1. We also show that the equidistribution phenomenon holds if the annulus of convergence degenerates to a circle. Moreover, equidistribution of zeros still persists when the Laurent rational functions diverge everywhere, which is new even in the deterministic case. All results hold under two types of general conditions on random coefficients. The first condition is that the random coefficients are non-trivial i.i.d. random variables with finite \(\log ^+\) moments. The second condition allows random variables that need not be independent or identically distributed, but only requires certain uniform bounds on the tails of their distributions. PubDate: 2017-11-01 DOI: 10.1007/s40315-017-0213-3

Authors:A. B. Bogatyrev; O. A. Grigor’ev Abstract: A new analytical method for the conformal mapping of rectangular polygons with a straight angle at infinity to a half-plane and back is proposed. The method is based on the observation that the SC integral in this case is an abelian integral on a hyperelliptic curve, so it may be represented in terms of Riemann theta functions. The approach is illustrated by the computation of 2D-flow of ideal fluid above rectangular underlying surface and the computation of the capacities of multi-component rectangular condensers with axial symmetry. PubDate: 2017-10-31 DOI: 10.1007/s40315-017-0217-z

Authors:Bartosz Łanucha Abstract: A truncated Toeplitz operator is a compression of the multiplicationoperator to a backward shift invariant subspace of the Hardy space \(H^2\) . Anasymmetric truncated Toeplitz operator is a compression of the multiplication operator that acts between two different backward shift invariant subspaces of \(H^2\) . All rank-one truncated Toeplitz operators have been described by Sarason. Here, we characterize all rank-one asymmetric truncated Toeplitz operators. This completes the description given by Łanucha for asymmetric truncated Toeplitz operators on finite-dimensional backward shift invariant subspaces. PubDate: 2017-10-30 DOI: 10.1007/s40315-017-0219-x

Authors:L. Bos; N. Levenberg Abstract: We prove a version of the Bernstein–Walsh theorem on uniform polynomial approximation of holomorphic functions on compact sets in several complex variables. Here we consider subclasses of the full polynomial space associated to a convex body P. As a consequence, we validate and clarify some observations of Trefethen in multivariate approximation theory. PubDate: 2017-10-24 DOI: 10.1007/s40315-017-0220-4

Authors:Kiran Kumar Behera; A. Swaminathan Abstract: The purpose of the present paper is to investigate some structural and qualitative aspects of two different perturbations of the parameters of g-fractions. In this context, the concept of gap g-fractions is introduced. While tail sequences of a continued fraction play a significant role in the first perturbation, Schur fractions are used in the second perturbation of the g-parameters that is considered. Illustrations are provided using Gaussian hypergeometric functions. Using a particular gap g-fraction, some members of the class of Pick functions are also identified. PubDate: 2017-10-24 DOI: 10.1007/s40315-017-0218-y

Authors:Nan Wu Abstract: Using the spread relation we investigate the growth of transcendental holomorphic curves when they have radially distributed small holomorphic curves. PubDate: 2017-06-26 DOI: 10.1007/s40315-017-0208-0

Authors:Jörg Liesen; Olivier Sète; Mohamed M. S. Nasser Abstract: We present a numerical method for computing the logarithmic capacity of compact subsets of \(\mathbb {C}\) , which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have any special symmetry. The method relies on the conformal map onto lemniscatic domains and, computationally, on the solution of a boundary integral equation with the Neumann kernel. Our numerical examples indicate that the method is fast and accurate. We apply it to give an estimate of the logarithmic capacity of the Cantor middle third set and generalizations of it. PubDate: 2017-06-22 DOI: 10.1007/s40315-017-0207-1