Authors:David Minda Pages: 579 - 590 Abstract: 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: 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: 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: 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: 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: 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: 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: 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: 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:Matthew Fleeman; Erik Lundberg Pages: 369 - 379 Abstract: Abstract Given a planar domain \(\Omega \) , the Bergman analytic content measures the \(L^{2}(\Omega )\) -distance between \(\bar{z}\) and the Bergman space \(A^{2}(\Omega )\) . We compute the Bergman analytic content of simply connected quadrature domains with quadrature formula supported at one point, and we also determine the function \(f \in A^2(\Omega )\) that best approximates \(\bar{z}\) . We show that, for simply connected domains, the square of the Bergman analytic content is equivalent to torsional rigidity from classical elasticity theory, while for multiply connected domains these two domain constants are not equivalent in general. PubDate: 2017-09-01 DOI: 10.1007/s40315-016-0189-4 Issue No:Vol. 17, No. 3 (2017)

Authors:Anders Gustafsson Pages: 381 - 394 Abstract: Abstract Let D denote a Dini domain in \(\mathbb {C}\) and let \(\overline{\mathbb {C}}=\mathbb {C}\cup \{\infty \}\) . For each \(n=1,2,3\ldots \) , take points \(A_n=\{a_{ni}\}_{i=0}^n\) in D and points \(B_n=\{b_{ni}\}_{i=1}^n\) in \(\overline{\mathbb {C}}{\setminus } D\) . Let \(\alpha _n\) and \(\beta _n\) be the normalized point counting measures of \(A_n\) and \(B_n\) . Suppose that \(\alpha _n\xrightarrow {w^*}\alpha \) , \(\beta _n\xrightarrow {w^*}\beta \) and denote by \(\alpha '\) and \(\beta '\) their swept measures onto \(\partial D\) . Denote by \(U_\mu \) the logarithmic potential of the measure \(\mu \) . We show that if \(\alpha '=\beta '\) and if \(\{(n+1)(U_{\alpha _n}-U_\alpha )\}\) , \(\{n(U_{\beta _n'}-U_{\beta '})\}\) uniformly have at most logarithmic growth at \(\partial D\) , then for every \(f\in A^{-\infty }(D)\) and for the rational interpolants \(r_{n,f}\) of degree n with poles at \(B_n\) interpolating to f at \(A_n\) , we have \(r_{n,f}\rightarrow f\) in \(A^{-\infty }(D)\) . PubDate: 2017-09-01 DOI: 10.1007/s40315-016-0187-6 Issue No:Vol. 17, No. 3 (2017)

Authors:Feng Lü; Weiran Lü Pages: 395 - 403 Abstract: Abstract In this work, we focus on a conjecture on the uniqueness problem of meromorphic functions sharing three distinct values with their difference operators, which is mentioned in Chen and Yi (Results Math 63, 557–565, 2013). We prove that the conjecture is true for meromorphic functions of finite order. Furthermore, a result of Zhang and Liao (Sci China Math 57, 2143–2152, 2014) is generalized from entire functions to meromorphic functions. PubDate: 2017-09-01 DOI: 10.1007/s40315-016-0188-5 Issue No:Vol. 17, No. 3 (2017)

Authors:Federico Piazzon Pages: 405 - 443 Abstract: Abstract The Bernstein–Markov property is an asymptotic quantitative assumption on the growth of uniform norms of polynomials or rational functions on a compact set with respect to \(L^2_\mu \) -norms, where \(\mu \) is a positive finite measure. We consider two variants of the Bernstein–Markov property for rational functions with restricted poles and compare them with the polynomial Bernstein–Markov property to find some sufficient conditions for the latter to imply the former. Moreover, we recover a sufficient mass-density condition for a measure to satisfy the rational Bernstein–Markov property on its support. Finally we present, as an application, a meromorphic \(L^2\) version of the Bernstein–Walsh Lemma. PubDate: 2017-09-01 DOI: 10.1007/s40315-017-0194-2 Issue No:Vol. 17, No. 3 (2017)

Authors:Jochen Becker; Christian Pommerenke Pages: 487 - 497 Abstract: Abstract Let the function \(\varphi \) be holomorphic in the unit disk \(\mathbb {D}\) and let \(\varphi (\mathbb {D})\subset \mathbb {D}\) . We consider points \(\zeta \in \partial \mathbb {D}\) where \(\varphi \) has an angular limit \(\varphi (\zeta )\in \partial \mathbb {D}\) and study the behaviour of \((\varphi (z)-\varphi (\zeta ))/(z-\zeta )\) as z tends to \(\zeta \) in various ways. In particular, there is a result connecting \( \varphi '(\zeta _{\nu }) \) and \( \varphi (\zeta _{\mu })-\varphi (\zeta _{\nu }) \) for three points \(\zeta _{\nu }\) . Expressed as a positive semidefinite quadratic form, this result could, perhaps, be generalized to n points \(\zeta _{\nu }\) . PubDate: 2017-09-01 DOI: 10.1007/s40315-017-0199-x Issue No:Vol. 17, No. 3 (2017)

Authors:Raymond Cheng; Javad Mashreghi; William T. Ross Pages: 499 - 523 Abstract: Abstract Bounds are obtained for the zeros of an analytic function on a disk in terms of the Taylor coefficients of the function. These results are derived using the notion of Birkhoff–James orthogonality in the sequence space \(\ell ^p\) with \(p \in (1, \infty )\) , along with an associated Pythagorean theorem. It is shown that these methods are able to reproduce, and in some cases sharpen, some classical bounds for the roots of a polynomial. PubDate: 2017-09-01 DOI: 10.1007/s40315-017-0191-5 Issue No:Vol. 17, No. 3 (2017)

Authors:Simon Baker; Han Yu Abstract: Abstract Given \(H\subseteq \mathbb {C}\) two natural objects to study are the set of zeros of polynomials with coefficients in H, $$\begin{aligned} \left\{ z\in \mathbb {C}: \exists k>0,\, \exists (a_n)\in H^{k+1}, \sum _{n=0}^{k}a_{n}z^n=0\right\} , \end{aligned}$$ and the set of zeros of a power series with coefficients in H, $$\begin{aligned} \left\{ z\in \mathbb {C}: \exists (a_n)\in H^{\mathbb {N}}, \sum _{n=0}^{\infty } a_nz^n=0\right\} . \end{aligned}$$ In this paper, we consider the case where each element of H has modulus 1. The main result of this paper states that for any \(r\in (1/2,1),\) if H is \(2\cos ^{-1}(\frac{5-4 r ^2}{4})\) -dense in \(S^1,\) then the set of zeros of polynomials with coefficients in H is dense in \(\{z\in {\mathbb {C}}: z \in [r,r^{-1}]\},\) and the set of zeros of power series with coefficients in H contains the annulus \(\{z\in \mathbb {C}: z \in [r,1)\}\) . These two statements demonstrate quantitatively how the set of polynomial zeros/power series zeros fill out the natural annulus containing them as H becomes progressively more dense. PubDate: 2017-10-09 DOI: 10.1007/s40315-017-0215-1

Authors:Indrajit Lahiri; Shubhashish Das Abstract: Abstract In the paper we prove a uniqueness theorem for meromorphic functions that share a function of slower growth with linear differential polynomials. Our result is closely related to a conjecture of Brück. PubDate: 2017-10-05 DOI: 10.1007/s40315-017-0214-2

Authors:Nan Wu Abstract: 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: 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