Authors:Ming Li; Toshiyuki Sugawa Pages: 179 - 193 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: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

Authors:Ilgiz R Kayumov; Saminathan Ponnusamy 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: 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: 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: 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: 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

Authors:Jerry R. Muir 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: 2017-05-10 DOI: 10.1007/s40315-017-0203-5

Authors:Masayo Fujimura 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: 2017-05-04 DOI: 10.1007/s40315-017-0201-7

Authors:Thi Hoai An Ta; Viet Phuong Nguyen 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: 2017-04-25 DOI: 10.1007/s40315-017-0198-y

Authors:Jochen Becker; Christian Pommerenke 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: 2017-04-24 DOI: 10.1007/s40315-017-0199-x