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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: 2021-10-09
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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: 2021-10-01
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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: 2021-09-29
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Abstract: We present a corrected proof of Proposition 2.3 in the article “Boundary integral formula for harmonic functions on Riemann surfaces” [2]. PubDate: 2021-09-29
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Abstract: We consider transcendental meromorphic functions for which the zeros, 1-points and poles are distributed on three distinct rays. We show that such functions exist if and only if the rays are equally spaced. We also obtain a normal family analogue of this result. PubDate: 2021-09-22
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Abstract: Let M be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets D in M for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in \(L^2(D)\) , and the supremum of the torsion function for D are comparable with the square of the capacitary width for D if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale. PubDate: 2021-09-08
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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: 2021-09-03
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Abstract: In the present paper, we study the order of convexity of \(z{_2F_1}(a,b;c;z)\) with real parameters a, b and c where \({_2F_1}(a,b;c;z)\) is the Gaussian hypergeometric function. First we obtain some conditions for \(z{_2F_1}(a,b;c;z)\) with no finite orders of convexity by considering its asymptotic behavior around \(z=1\) . Then the order of convexity of \(z{_2F_1}(a,b;c;z)\) is demonstrated for some ranges of real parameters a, b and c. In the last section, we give some examples as applications of the main results. PubDate: 2021-09-01
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Abstract: In the article, we present the best possible bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals of the first and second kinds, which are the generalizations of previous results for complete p-elliptic integrals. PubDate: 2021-09-01
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Abstract: Given two harmonic functions \(f_{1}=h_{1}+{\bar{g}}_{1}\) and \(f_{2}=h_{2}+{\bar{g}}_{2}\) defined on the open unit disk of the complex plane, the geometric properties of the product \(f_{1}\otimes f_{2}\) defined by $$\begin{aligned} f_{1}\otimes f_{2}=h_{1}*h_{2}-\overline{g_{1}*g_{2}}+ i{\text {Im}}(h_{1}*g_{2}+h_{2}*g_{1}) \end{aligned}$$ are discussed. Here \(*\) denotes the analytic convolution. Sufficient conditions are obtained for the product to be univalent and convex in the direction of the real axis. In addition, a convolution theorem, coefficient inequalities and closure properties for the product \(\otimes \) are proved. PubDate: 2021-09-01
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Abstract: In the paper we consider the uniqueness problem of an entire function f when it shares a doubleton with its derivative \(f^{(k)}\) , \(k \ge 1\) . Our result extends a result of Li and Yang (J Math Soc Japan 51(4):781–799, 1999). PubDate: 2021-09-01
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Abstract: Let X be a compact subset of the complex plane with the property that every relatively open subset of X has positive area and let \(A_0(X)\) denote the space of VMO functions that are analytic on X. \(A_0(X)\) is said to admit a bounded point derivation of order t at a point \(x_0 \in \partial X\) if there exists a constant C such that \( f^{(t)}(x_0) \le C \Vert f\Vert _{{\text {BMO}}}\) for all functions in \({\text {VMO}}(X)\) that are analytic on X. In this paper, we give necessary and sufficient conditions in terms of lower 1-dimensional Hausdorff content for \(A_0(X)\) to admit a bounded point derivation at \(x_0\) . These conditions are similar to conditions for the existence of bounded point derivations on other functions spaces. PubDate: 2021-09-01
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Abstract: In this paper, we study the unicity of entire functions concerning their shifts and derivatives and prove: Let f be a non-constant entire function of hyper-order less than 1, let c be a non-zero finite value, and let a, b be two distinct finite values. If \(f'(z)\) and \(f(z+c)\) share a, b IM, then \(f'(z)\equiv f(z+c)\) . This improves some results due to Qi and Yang (Comput Methods Funct Theory 20:159–178, 2020). PubDate: 2021-09-01
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Abstract: In this paper, by using the formula of Faà di Bruno for the k-th derivative of a composite function, we establish a refinement of the Fekete and Szegő inequality for a class of holomorphic mappings on bounded starlike circular domains in \(\mathbb {C}^n\) . The results presented here generalize some known results. Finally, a certain problem is also proposed. PubDate: 2021-09-01
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Abstract: We study questions of existence and uniqueness of quadrature domains using computational tools from real algebraic geometry. These problems are transformed into questions about the number of solutions to an associated real semi-algebraic system, which is analyzed using the method of real comprehensive triangular decomposition. PubDate: 2021-09-01
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Abstract: We improve a normality result of Liu–Li–Pang [4] concerning shared values between two families. Let \(\mathcal F\) and \(\mathcal G\) be two families of meromorphic functions on D whose zeros are multiple. Suppose that \(\mathcal G\) is normal on D, and no sequence contained in \(\mathcal G\) \(\chi \) -converges locally uniformly to \(\infty \) or a function g satisfying \(g'\equiv 1\) . If for every \(f\in \mathcal F\) , there exists a function \(g\in \mathcal G\) such that f and g share 0 and \(\infty \) while \(f'\) and \(g'\) share 1, then \(\mathcal F\) is also normal on D. PubDate: 2021-09-01
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Abstract: The van der Pauw method is a well-known experimental technique in the applied sciences for measuring physical quantities such as the electrical conductivity or the Hall coefficient of a given sample. Its popularity is attributable to its flexibility: the same method works for planar samples of any shape provided they are simply connected. Mathematically, the method is based on the cross-ratio identity. Much recent work has been done by applied scientists attempting to extend the van der Pauw method to samples with holes (“holey samples”). In this article we show the relevance of two new function theoretic ingredients to this area of application: the prime function associated with the Schottky double of a multiply connected planar domain and the Fay trisecant identity involving that prime function. We focus here on the single-hole (doubly connected, or genus one) case. Using these new theoretical ingredients we are able to prove several mathematical conjectures put forward in the applied science literature. PubDate: 2021-08-31
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Abstract: Let \(\Omega \) be a smooth, bounded, convex domain in \({\mathbb {R}}^n\) and let \(\Lambda _k\) be a finite subset of \(\Omega \) . We find necessary geometric conditions for \(\Lambda _k\) to be interpolating for the space of multivariate polynomials of degree at most k. Our results are asymptotic in k. The density conditions obtained match precisely the necessary geometric conditions that sampling sets are known to satisfy and are expressed in terms of the equilibrium potential of the convex set. Moreover we prove that in the particular case of the unit ball, for k large enough, there are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most k. PubDate: 2021-08-30
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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: 2021-08-26
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