Authors:Manas Ranjan Mohapatra; Swadesh Kumar Sahoo Abstract: We mainly consider two metrics: a Gromov hyperbolic metric and a scale-invariant Cassinian metric. We compare these two metrics and obtain their relationship with certain well-known hyperbolic-type metrics, leading to several inclusion relations between the associated metric balls. PubDate: 2018-02-07 DOI: 10.1007/s40315-018-0233-7

Authors:Gunter Semmler; Elias Wegert Abstract: The phase plot of the function depicted on the cover of this volume is doubly periodic. In this expository paper, we discuss a canonical representation of all functions with doubly periodic phase (argument) in terms of the Weierstrass \(\sigma \) -function. In particular, we point out that the zeros and poles of such a function in a fundamental domain can be prescribed arbitrarily, with the only restriction that their total numbers (counting multiplicities) must coincide. PubDate: 2018-02-03 DOI: 10.1007/s40315-018-0236-4

Authors:Jörg Liesen; Jan Zur Abstract: Generalizing several previous results in the literature on rational harmonic functions, we derive bounds on the maximum number of zeros of functions \(f(z) = \frac{p(z)}{q(z)} - \overline{z}\) , which depend on both \(\mathrm{deg}(p)\) and \(\mathrm{deg}(q)\) . Furthermore, we prove that any function that attains one of these upper bounds is regular. PubDate: 2018-01-25 DOI: 10.1007/s40315-017-0231-1

Authors:Timothy Ferguson Abstract: We discuss approximation of extremal functions by polynomials in the weighted Bergman spaces \(A^p_\alpha \) where \(-1< \alpha < \min (0,p-2)\) . We obtain bounds on how close the approximation is to the true extremal function in the \(A^p_\alpha \) and uniform norms. We also prove several results on the relation between the Bergman modulus of continuity of a function and how quickly its best polynomial approximants converge to it. PubDate: 2018-01-24 DOI: 10.1007/s40315-017-0230-2

Authors:Ruishen Qian; Songxiao Li Abstract: Under some mild conditions on the weight function \(\rho \) , we characterize lacunary series in Dirichlet-type spaces. Moreover, we also obtain a new characterization of Dirichlet-type spaces in terms of pseudoanalytic extension. Two applications are also given. PubDate: 2018-01-16 DOI: 10.1007/s40315-017-0228-9

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:Oh Sang Kwon; Adam Lecko; Young Jae Sim Abstract: In the present paper, a formula for the fourth coefficient of Carathéodory functions was computed. PubDate: 2017-12-18 DOI: 10.1007/s40315-017-0229-8

Authors:Mark Elin; David Shoikhet; Nikolai Tarkhanov Abstract: In this manuscript we provide a review on the classical and resent results related to the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators. PubDate: 2017-12-08 DOI: 10.1007/s40315-017-0227-x

Authors:David Kalaj; Elver Bajrami Abstract: We prove some isoperimetric type inequalities for real harmonic functions in the unit disk belonging to the Hardy space \(h^p\) , \(p>1\) , and for complex harmonic functions in \(h^4\) . The results extend some recent results in the area. Further, we discuss some Riesz type results for holomorphic functions. PubDate: 2017-12-04 DOI: 10.1007/s40315-017-0226-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