Authors:Andrea del Monaco; Ikkei Hotta; Sebastian Schleißinger Pages: 9 - 33 Abstract: In this note, we consider a multi-slit Loewner equation with constant coefficients that describes the growth of multiple SLE curves connecting N points on \(\mathbb {R}\) to infinity within the upper half-plane. For every \(N\in \mathbb {N}\) , this equation yields a measure-valued process \(t\mapsto \{\alpha _{N,t}\},\) and we are interested in the limit behaviour as \(N\rightarrow \infty .\) We prove tightness of the sequence \(\{\alpha _{N,t}\}_{N\in \mathbb {N}}\) under certain assumptions and address some further problems. Moreover, we investigate a similar situation in which all slits are trajectories of a certain quadratic differential. PubDate: 2018-03-01 DOI: 10.1007/s40315-017-0205-3 Issue No:Vol. 18, No. 1 (2018)

Authors:Anton Bohdanov Pages: 35 - 51 Abstract: We give explicit values of the parameters \(a>1\) and \(m \in (0,1)\) for which an entire function \(f^{(m,a)}(z)=\sum _{k=0}^\infty \frac{z^k}{a^{k^2}}(k!)^{m}\) belongs to the Laguerre–Pólya class. We also consider the values of parameters for which the Taylor sections of this function belong to the Laguerre–Pólya class. PubDate: 2018-03-01 DOI: 10.1007/s40315-017-0210-6 Issue No:Vol. 18, No. 1 (2018)

Authors:Raymond Mortini; Rudolf Rupp Pages: 53 - 87 Abstract: It is well known that in the disk algebra \(A({ \mathbb D})\) every zero-free function has a logarithm in \(A({ \mathbb D})\) . This is no longer true if we look at invertible matrices over \(A({ \mathbb D})\) . In this paper, we give a sufficient condition on the trace of a \(2\times 2\) -matrix M so that \(M=e^L\) for some matrix \(L\in A({ \mathbb D})\) . We compute all the logarithms of the identity matrix in \({\mathcal M}_2(A({ \mathbb D}))\) and observe that the anti-diagonal elements can be arbitrarily prescribed. We also characterize those upper (or lower) triangular matrices which are exponentials in \({\mathcal M}_2(A({ \mathbb D}))\) and determine all their logarithms. This will enable us to prove that \(\exp {\mathcal M}_2(A({ \mathbb D}))\) is neither closed nor open within the principal component of \({\mathcal M}_2(A({ \mathbb D}))^{-1}\) . Finally, we show that every invertible matrix in \({\mathcal M}_2(A({ \mathbb D}))\) is a product of four exponential matrices and give conditions for reducing this number. These results will be put into the more general setting of commutative Banach algebras whenever possible. PubDate: 2018-03-01 DOI: 10.1007/s40315-017-0209-z Issue No:Vol. 18, No. 1 (2018)

Authors:Simon Baker; Han Yu Pages: 89 - 97 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: 2018-03-01 DOI: 10.1007/s40315-017-0215-1 Issue No:Vol. 18, No. 1 (2018)

Authors:Nisha Bohra; V. Ravichandran Pages: 99 - 123 Abstract: For three different normalizations of Bessel functions of first kind, the radius of k-parabolic starlikeness and k-uniform convexity of order \(\alpha \) are determined. The radius of strong starlikeness and other related radius are also obtained for these functions. We also find optimal parameters for which these functions are k-parabolic starlike and k-uniformly convex in the open unit disk. PubDate: 2018-03-01 DOI: 10.1007/s40315-017-0216-0 Issue No:Vol. 18, No. 1 (2018)

Authors:Indrajit Lahiri; Shubhashish Das Pages: 125 - 142 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: 2018-03-01 DOI: 10.1007/s40315-017-0214-2 Issue No:Vol. 18, No. 1 (2018)

Authors:Igor E. Pritsker Pages: 143 - 157 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: 2018-03-01 DOI: 10.1007/s40315-017-0213-3 Issue No:Vol. 18, No. 1 (2018)

Authors:Wensheng Cao; John R. Parker Pages: 159 - 191 Abstract: We give a generalisation of Shimizu’s lemma to complex or quaternionic hyperbolic space in any dimension for groups of isometries containing an arbitrary parabolic map. This completes a project begun by Kamiya (Hiroshima Math J 13:501–506, 1983). It generalises earlier work of Kamiya, Inkang Kim and Parker. The analogous result for real hyperbolic space is due to Waterman (Adv Math 101:87–113, 1993). PubDate: 2018-03-01 DOI: 10.1007/s40315-017-0212-4 Issue No:Vol. 18, No. 1 (2018)

Authors:Jian-Lin Li Abstract: Let \(f(z)=\sum _{n=1}^{\infty }a_{n}z^{n}\) be a conformal mapping of the unit disk \({\mathbb {D}}\) onto a domain starlike with respect to the origin. In this note, we prove that if \(g(z)=\sum _{n=1}^{\infty }b_{n}z^{n}\) is analytic in \({\mathbb {D}}\) and is subordinate to f(z); then, some coefficient inequalities involving \(a_{n}\) and \(b_{n}\) hold for all n. The results here extend the known results in a reasonable manner. PubDate: 2018-02-08 DOI: 10.1007/s40315-018-0232-8

Authors:Masahiko Taniguchi Abstract: Let D be a Koebe circle domain in the complex plane of infinite type. We show that, when D is tame, D corresponds to an infinitely generated extended classical Schottky group G and the quasiconformal Koebe space of D can be identified with the quasiconformal deformation space of G. Moreover, when D also satisfies the boundedness conditions, we prove that the essential Teichmüller space of the G-orbit of \(\infty \) admits the complex analytic structure. PubDate: 2018-02-08 DOI: 10.1007/s40315-018-0235-5

Authors:Vladimir Andrievskii Abstract: We establish sharp \(L_p\) , \(1\le p<\infty \) , weighted Remez- and Nikolskii-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) curve in the complex plane. PubDate: 2018-02-08 DOI: 10.1007/s40315-018-0234-6

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: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