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Abstract: Abstract In this paper we review the familiar collection of results that concern holomorphic maps of a disc or half-plane into itself that are due to Schwarz, Pick, Julia, Denjoy and Wolff. We give a coherent geometric treatment of these results entirely in terms of the ideas of geodesics, horocycles and G-spaces as introduced by Busemann. In particular, we show that the results of Wolff and Julia hold for all weak contractions of the hyperbolic metric (whether holomorphic or not); holomorphicity plays no role in the arguments. These results apply to holomorphic maps because the Schwarz–Pick lemma implies that holomorphic maps are weak contractions. An important ingredient in the proofs are several projections of the hyperbolic plane onto a geodesic which are weak contractions relative to the hyperbolic distance. PubDate: 2022-12-23

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Abstract: Abstract Let \(D\ne \mathbb {C}\) be a simply connected domain and f be a Riemann mapping from \(\mathbb {D}\) onto D. The Hardy number of D is the supremum of all p for which f belongs in the Hardy space \({H^p}\left( \mathbb {D} \right) \) . A comb domain is a domain whose complement is the union of an infinite number of vertical rays symmetric with respect to the real axis. In this paper we prove that, for \(p>0\) , there is a comb domain with Hardy number equal to p if and only if \(p\in [1,+\infty ]\) . It is known that the Hardy number is related to the moments of the exit time of Brownian motion from the domain. In fact, Burkholder proved that the Hardy number of a simply connected domain is twice the supremum of all \(p>0\) for which the p-th moment of the exit time of Brownian motion is finite. Therefore, our result implies that given \( p < q\) there exists a comb domain with finite p-th moment but infinite q-th moment if and only if \(q\ge 1/2\) . This answers a question posed by Boudabra and Markowsky. PubDate: 2022-12-01

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Abstract: Abstract We give an in-depth analysis of a 1-parameter family of electrified droplets first described in [19]. We also investigate a technique for searching for new solutions to the droplet equation, and rederive via this technique a 1 parameter family of physical droplets, which were first discovered by Crowdy [4]. We speculate on extensions of these solutions, in particular to the case of a droplet with multiple connected components. PubDate: 2022-12-01

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Abstract: Abstract In the present paper, we establish geometric properties, such as starlikeness and convexity of order \(\alpha \) ( \(0 \le \alpha < 1\) ), and close-to-convexity in the open unit disk \({\mathbb {U}} := \{z \in {\mathbb {C}}: z <1\}\) for a combination of a normalized form of the generalized Struve function of order p, \(w_{p,b,c}(z)\) , defined by \(D_{p,b,c}(z) = 2^{p}\sqrt{\pi } \, \Gamma (p + b/2 + 1) z^{(-p+1)/2}d_{p,b,c}(\sqrt{z})\) , where \(d_{p,b,c}(z) := -pw_{p,b,c}(z)+zw_{p,b,c}^{\prime }(z)\) , with \(p, b, c \in {\mathbb {C}}\) and \(\kappa = p + b/2+1 \notin \{0,-1,-2,\dots \}\) . We determine conditions for the parameters c and \(\kappa \) for which \(f \in {\mathcal {R}}(\beta ) = \left\{ f \in {\mathcal {A}}({\mathbb {U}}): \mathrm{Re} f^{\prime }(z) > \beta , z \in {\mathbb {U}} \right\} \) , \(0 \le \beta <1\) , indicates that the convolution product \(D_{p,b,c}*f\) belongs to the spaces \({\mathcal {H}}^{\infty }({\mathbb {U}})\) and \({\mathcal {R}}(\gamma )\) with \(\gamma \) depending on \(\alpha \) and \(\beta \) , where \({\mathcal {A}}({\mathbb {U}})\) denotes the class of all normalized analytic functions in \({\mathbb {U}}\) and \({\mathcal {H}}^{\infty }({\mathbb {U}})\) is the space of all bounded analytic functions in \({\mathcal {A}}({\mathbb {U}})\) . We also obtain sufficient conditions in terms of the expansion coefficients for \(f \in {\mathcal {A}}({\mathbb {U}})\) to be in some subclasses of the class of univalent functions. Motivation has come from the vital role of special functions in geometric function theory. PubDate: 2022-12-01

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Abstract: Abstract We consider translation and dilation mappings acting on the spaces of meromorphic functions on the complex plane and the punctured complex plane, respectively. In both cases, we show that there is a dense \(G_{\delta }\) -subset of meromorphic functions that are common universal for certain uncountable families of these mappings. While a corresponding result for translations exists for entire functions, our result for dilations has no holomorphic counterpart. We further obtain an analogue of Ansari’s Theorem for the mappings we consider, which is used as a key tool in the proofs of our main results. PubDate: 2022-12-01

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Abstract: Abstract The theory of multidimensional quasiconformal mappings employs three main approaches: analytic, geometric (modulus) and metric ones. In this paper, we use the last approach and establish the relationship between homeomorphisms of finite metric distortion (FMD-homeomorphisms), finitely bi-Lipschitz, quasisymmetric and quasiconformal mappings on Riemannian manifolds. One of the main results shows that FMD-homeomorphisms are lower Q-homeomorphisms. As an application, there are obtained some sufficient conditions for boundary extensions of FMD-homeomorphisms. These conditions are illustrated by several examples of FMD-homeomorphisms. PubDate: 2022-12-01

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Abstract: Abstract We consider weighted uniform convergence of entire analogues of the Grünwald operator on the real line. The main result deals with convergence of entire interpolations of exponential type \(\tau >0\) at zeros of Bessel functions in spaces with homogeneous weights. We discuss extensions to Grünwald operators from de Branges spaces. PubDate: 2022-12-01

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Abstract: Abstract Let G be a finite Jordan domain bounded by a Dini-smooth curve \(\Gamma \) in the complex plane \({\mathbb {C}}\) . In this work, approximation properties of the Faber–Laurent rational series expansions in variable exponent Morrey spaces \(L^{p(\cdot ),\lambda (\cdot )}(\Gamma )\) are studied. Also, direct theorems of approximation theory in variable exponent Morrey–Smirnov classes, defined in domains with a Dini-smooth boundary, are proved. PubDate: 2022-12-01

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Abstract: Abstract The boundedness of the difference of composition operators acting from the analytic Besov spaces to the Bloch type spaces is characterized. Some upper and lower bounds for the essential norm of the operator are also given. PubDate: 2022-12-01

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Abstract: Abstract In this paper, we study uniqueness questions for meromorphic functions for which certain difference polynomials share a finite non-zero value, and give mathematical expressions for the meromorphic functions in the conclusions of the main results in the present paper, which are the related to the questions studied in Li–Yu (Bull Korean Math Soc 55(5):1529–1561, 2018). PubDate: 2022-12-01

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Abstract: Abstract Given an analytic function g and a \(\mathcal {D}\) weight \(\omega \) on the unit disk \(\mathbb {D}=\{z \in \mathbb {C} : z <1\}\) , we characterize the boundedness and compactness of the Volterra-type integration operator $$\begin{aligned} J_{g}(f)(z)=\int _{0}^{z}f(\lambda )g'(\lambda )d\lambda \end{aligned}$$ between the weighted Bergman spaces \(L_{a}^{p}(\omega )\) and the Hardy spaces \(H^{q}\) for \(0<p,q<\infty \) . PubDate: 2022-10-21 DOI: 10.1007/s40315-022-00474-0

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Abstract: Abstract In this article we consider Paatero’s classes V(k) of functions of bounded boundary rotation as subsets of the Hornich space \(\mathcal H\) . We show that for a fixed \(k\ge 2\) the set V(k) is a closed and convex subset of \(\mathcal H\) and is not compact. We identify the extreme points of V(k) in \(\mathcal H\) . PubDate: 2022-09-20 DOI: 10.1007/s40315-022-00472-2

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Abstract: Abstract We deal with the functional inequality $$\begin{aligned} f(x)f(y) - f(xy) \le f (x) + f (y) - f(x+y) \end{aligned}$$ for \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) , which was introduced by Horst Alzer and Luis Salinas. We show that if f is a solution that is differentiable at 0 and \(f(0)=0\) , then \(f=0\) on \({\mathbb {R}}\) or \(f(x) = x\) for all \(x \in {\mathbb {R}}\) . Next, we prove that every solution f which satisfies some mild regularity and such that \(f(0)\ne 0\) is globally bounded. PubDate: 2022-09-16 DOI: 10.1007/s40315-022-00473-1

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Abstract: Abstract For \(0<p \le \infty \) , let \(H^p\) denote the classical Hardy space of the unit disc. We consider the extremal problem of maximizing the modulus of the kth Taylor coefficient of a function \(f \in H^p\) which satisfies \(\Vert f\Vert _{H^p}\le 1\) and \(f(0)=t\) for some \(0 \le t \le 1\) . In particular, we provide a complete solution to this problem for \(k=1\) and \(0<p<1\) . We also study F. Wiener’s trick, which plays a crucial role in various coefficient-related extremal problems for Hardy spaces. PubDate: 2022-09-12 DOI: 10.1007/s40315-022-00469-x

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Abstract: Abstract In this paper we define new subclasses of univalent mappings in the case of several complex variables. We will focus our attention on a particular class, denoted \(E^*_1\) , and observe that in the case of one complex variable, \(E^*_1(U)\) coincides with the class of convex functions K on the unit disc. However, if \(n\ge {2}\) , then \(E^*_1(\mathbb {B}^n)\) is different from the class of convex mappings \(K(\mathbb {B}^n)\) on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\) . Along with this, we will study other properties of the class \(E^*_1\) on the unit polydisc, respectively on the Euclidean unit ball in \(\mathbb {C}^n\) . In the second part of the paper we discuss the Graham–Kohr extension operator \(\Psi _{n,\alpha }\) (defined by Graham and Kohr in Complex Variab. Theory Appl. 47:59–72, 2002). They proved that the extension operator \(\Psi _{n,\alpha }\) does not preserve convexity for \(n\ge {2}\) for all \(\alpha \in [0,1]\) . However, in this paper we prove that \(\Psi _{n,0}(K)\) and \(\Psi _{n,1}(K)\) are subsets of the class \(E^*_1(\mathbb {B}^n)\) which is different from the class \(K(\mathbb {B}^n)\) for the Euclidean case. PubDate: 2022-09-09 DOI: 10.1007/s40315-022-00467-z

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Abstract: Abstract Comparing phase plots of truncated series solutions of Kepler’s equation by Lagrange’s power series with those by Bessel’s Kapteyn series strongly suggests that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence in the complex plane is a cluster point of zeros of sections of the series. We prove this result by studying properties of the growth function of a sequence of entire functions. For series, this growth function is computable in terms of the convergence abscissa of an associated general Dirichlet series. The proof then extends, besides including Jentzsch’s classical result for power series, to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel functions. Moreover, sections of Kapteyn and Neumann series generally exhibit zeros close to the real axis which can be explained, including their asymptotic linear density, by the theory of the distribution of zeros of entire functions. PubDate: 2022-09-09 DOI: 10.1007/s40315-022-00468-y

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Abstract: Abstract We introduce a concept of a quasi proximate order which is a generalization of a proximate order and allows us to study efficiently analytic functions whose order and lower order of growth are different. We prove an existence theorem for a quasi proximate order, i.e. a counterpart of Valiron’s theorem for a proximate order. As applications, we generalize and complement some results of M. Cartwright and C. N. Linden on asymptotic behavior of analytic functions in the unit disc. PubDate: 2022-09-01 DOI: 10.1007/s40315-021-00411-7

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Abstract: Abstract For a domain \( D \subsetneq {\mathbb {R}}^{n} \) , Ibragimov’s metric is defined as $$\begin{aligned} u_{D}(x,y) = 2\, \log \frac{ x-y +\max \{d(x),d(y)\}}{\sqrt{d(x)\,d(y)}}, \quad \quad x,y \in D, \end{aligned}$$ where d(x) denotes the Euclidean distance from x to the boundary of D. In this paper, we compare Ibragimov’s metric with the classical hyperbolic metric in the unit ball or in the upper half space, and prove sharp comparison inequalities between Ibragimov’s metric and some hyperbolic type metrics. We also obtain several sharp distortion inequalities for Ibragimov’s metric under some families of Möbius transformations. PubDate: 2022-09-01 DOI: 10.1007/s40315-021-00414-4