Authors:Ming Li; Toshiyuki Sugawa Pages: 179 - 193 Abstract: Abstract In this note, we investigate the supremum and the infimum of the functional \( a_{n+1} - a_{n} \) for functions, convex and analytic on the unit disk, of the form \(f(z)=z+a_2z^2+a_3z^3+\cdots .\) We also consider the related problem of maximizing the functional \( a_{n+1}-a_{n} \) for convex functions f with \(f''(0)=p\) for a prescribed \(p\in [0,2].\) PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0177-8 Issue No:Vol. 17, No. 2 (2017)

Authors:Gary G. Gundersen Pages: 195 - 209 Abstract: Abstract Twenty-eight research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone. PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0178-7 Issue No:Vol. 17, No. 2 (2017)

Authors:Eusebio Ariza García; Antonio Di Teodoro; María Sapiain; Franklin Vargas Pages: 211 - 236 Abstract: Abstract Consider the initial value problem 0.1 $$\begin{aligned} \partial _{t}u= & {} {\mathcal L}(t,x,u,\partial _{x_{i}}u),\nonumber \\ u(0,x)= & {} \varphi (x), \end{aligned}$$ where t is the time, \({\mathcal L}\) is a linear first-order differential operator and \(\varphi \) is a generalized q-metamonogenic function. This problem can be solved by applying the method of associated spaces which is constructed by Tutschke (see Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer, New York, 1989). In this work, we formulate sufficient conditions on the coefficients of the operator \({\mathcal L}\) under which this operator is associated to the space of generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side, when q and \(\lambda \) are constant Clifford vectors. We also build a computational algorithm to check the computations in the cases \({\mathcal A}^{*}_{2,2}\) and \({\mathcal A}^{*}_{3,2}\) . In conical domains, the initial value problem (0.1) is uniquely solvable for an operator \({\mathcal L}\) and for any generalized q-metamonogenic initial function \(\varphi \) , provided an interior estimate holds for generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side. The solution is also a generalized q-metamonogenic function for each fixed t. This work generalizes the results given in Di Teodoro and Sapian (Adv. Appl. Clifford Algebras, 25:283–301, 2015) and Van (Differential operator in a Clifford analysis associated to differential equations with anti-monogenic right hand side, IC/2006/134, 2016). PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0182-y Issue No:Vol. 17, No. 2 (2017)

Authors:Ekin Uğurlu; Elgiz Bairamov Pages: 237 - 253 Abstract: Abstract In this paper, we introduce a new approach for the spectral analysis of a linear second order dissipative differential operator with distributional potentials. This approach is related with the inverse operator. We show that the inverse operator is a non-selfadjoint trace class operator. Using Lidskiĭ’s theorem, we introduce a complete spectral analysis of the second order dissipative differential operator. Moreover, we give a trace formula for the trace class integral operator. PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0185-8 Issue No:Vol. 17, No. 2 (2017)

Authors:Marc Technau; Niclas Technau Pages: 255 - 272 Abstract: Abstract It is well-known that the growth of a slit in the upper half-plane can be encoded via the chordal Loewner equation, which is a differential equation for schlicht functions with a certain normalisation. We prove that a multiple slit Loewner equation can be used to encode the growth of the union \(\Gamma \) of multiple slits in the upper half-plane if the slits have pairwise disjoint closures. Under certain assumptions on the geometry of \(\Gamma \) , our approach allows us to derive a Loewner equation for infinitely many slits as well. PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0179-6 Issue No:Vol. 17, No. 2 (2017)

Authors:Juan Arango; Hugo Arbeláez; Diego Mejía Pages: 273 - 288 Abstract: Abstract We present the notion of lower spherical order for locally injective meromorphic functions in the unit disk, and study some properties of functions with positive lower spherical order. PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0181-z Issue No:Vol. 17, No. 2 (2017)

Authors:ZhiHong Liu; Saminathan Ponnusamy Pages: 289 - 302 Abstract: Abstract We consider the convolution of half-plane harmonic mappings with respective dilatations \((z+a)/(1+az)\) and \(e^{i\theta }z^{n}\) , where \(-1<a<1\) and \(\theta \in \mathbb {R},n\in \mathbb {N}\) . We prove that such convolutions are locally univalent for \(n=1\) , which solves an open problem of Dorff et al. (see J Anal 18:69–81 [3, Problem 3.26]). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for \(n\ge 2\) . PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0180-0 Issue No:Vol. 17, No. 2 (2017)

Authors:Qinghua Hu; Songxiao Li; Yecheng Shi Pages: 303 - 318 Abstract: Abstract In this paper, we give a new characterization for the boundedness, compactness and essential norm of differences of weighted composition operators between weighted-type spaces. PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0184-9 Issue No:Vol. 17, No. 2 (2017)

Authors:Darren Crowdy Pages: 319 - 341 Abstract: Abstract A covering map formalism for studying the spectral curves associated with finite gap Jacobi matrices is presented. We advocate a constructive function theoretic framework based on use of the Schottky–Klein prime function. The single gap, or genus-one, case is studied in explicit detail. PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0186-7 Issue No:Vol. 17, No. 2 (2017)

Authors:D. B. Karp; E. G. Prilepkina Pages: 343 - 367 Abstract: Abstract In this paper, we find several new properties of a class of Fox’s H functions which we call delta neutral. In particular, we find an expansion in the neighborhood of the finite non-zero singularity and give new Mellin transform formulas under a special restriction on parameters. The last result is applied to prove a conjecture regarding the representing measure for gamma ratio in Bernstein’s theorem. Furthermore, we find the weak limit of measures expressed in terms of the H function which furnishes a regularization method for integrals containing the delta neutral and zero-balanced cases of Fox’s H function. We apply this result to extend a recently discovered integral equation to the zero-balanced case. In the last section of the paper, we consider a reduced form of this integral equation for Meijer’s G function. This leads to certain expansions believed to be new even in the case of the Gauss hypergeometric function. PubDate: 2017-06-01 DOI: 10.1007/s40315-016-0183-x Issue No:Vol. 17, No. 2 (2017)

Authors:Anton Bohdanov Abstract: 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: 2017-07-10 DOI: 10.1007/s40315-017-0210-6

Authors:Raymond Mortini; Rudolf Rupp Abstract: 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: 2017-06-29 DOI: 10.1007/s40315-017-0209-z

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

Authors:Ilgiz R Kayumov; Saminathan Ponnusamy 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: 2017-05-27 DOI: 10.1007/s40315-017-0206-2

Authors:Andrea del Monaco; Ikkei Hotta; Sebastian Schleißinger Abstract: 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: 2017-05-25 DOI: 10.1007/s40315-017-0205-3

Authors:Yuk-J. Leung 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: 2017-05-18 DOI: 10.1007/s40315-017-0204-4

Authors:José L. Fernández Abstract: Abstract We present a streamlined proof (and some refinements) of a characterization (due to F. Carlson and G. Bourion, and also to P. Erdős and H. Fried) of the so-called Szegő power series. This characterization is then applied to readily obtain some (more) recent known results and some new results on the asymptotic distribution of zeros of sections of random power series, extricating quite naturally the deterministic ingredients. Finally, we study the possible limits of the zero counting probabilities of a power series. PubDate: 2017-05-18 DOI: 10.1007/s40315-017-0200-8

Authors:Hu Chunying; Shi Qingtian 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: 2017-05-11 DOI: 10.1007/s40315-017-0202-6