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Abstract: Abstract Inspired by the questions Gundersen and Yang proposed, we investigate the exact forms of the entire solutions of the following two types of binomial differential equations $$\begin{aligned} a(z)ff''+b(z)(f')^2=c(z)e^{2q(z)}; \\ a(z)f'f''+b(z)f^2=c(z)e^{2q(z)}, \end{aligned}$$ where a, b, c are polynomials with no common zeros satisfying \(abc\not \equiv 0\) , and q is a non-constant polynomial. PubDate: 2024-08-09 DOI: 10.1007/s40315-024-00556-1
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Abstract: Abstract We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, \(n\ge 3\) . The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it ifollows that if \( \{ f_{n} \}_{n=1}^{\infty } \) is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion \(H({f_{n}})\) ), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower-semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower-semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence \( \{f_{n} \}_{n=1}^{\infty } \) with \( {f_{n}}\rightarrow {f}\) locally uniformly and with \(\limsup _{n\rightarrow \infty } H( {f_{n}})<H( {f})\) . Our main result shows this is true for affine mappings. Addressing conjectures of Gehring and Iwaniec we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each \(\alpha <\sqrt{2}\) there is \({f_{n}}\rightarrow {f}\) locally uniformly with f affine and $$\begin{aligned} \alpha \; \limsup _{n\rightarrow \infty } H( {f_{n}}) < H( {f}) \end{aligned}$$ We conjecture \(\sqrt{2}\) to be best possible. PubDate: 2024-08-07 DOI: 10.1007/s40315-024-00555-2
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Abstract: Abstract We study the linear dilatation of the mappings satisfying an inverse Poletsky inequality in metric spaces. We also show that under certain conditions such mappings are quasiregular. PubDate: 2024-07-03 DOI: 10.1007/s40315-024-00553-4
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Abstract: Abstract We provide an elementary derivation of the Bessel analog of the celebrated Riesz composition formula and use the former to effortlessly derive the latter. PubDate: 2024-07-02 DOI: 10.1007/s40315-024-00539-2
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Abstract: Abstract Diameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameters of a class of simple Riemann surfaces known as generalized Bolza surfaces of any genus greater than 1 are equal to the radii of their fundamental polygons. This is the first exact result for the diameter of a compact hyperbolic manifold. PubDate: 2024-06-27 DOI: 10.1007/s40315-024-00546-3
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Abstract: Abstract We show that the Poincaré inequality holds on an open set \(D\subset \mathbb {R}^n\) if and only if D admits a smooth, bounded function whose Laplacian has a positive lower bound on D. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on D is equivalent to the finiteness of the strict inradius of D measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet–Laplacian. PubDate: 2024-06-26 DOI: 10.1007/s40315-024-00550-7
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Abstract: Abstract In this article, we present the symmetry group of a global slice Dirac operator and its iterated ones. Further, the explicit forms of intertwining operators of the iterated global slice Dirac operator are given. At the end, we introduce a variant of the global slice Dirac operator, which allows functions considered to be defined on the whole Euclidean space. The invariance property and the intertwining operators of this variant of the global slice Dirac operator are also presented. PubDate: 2024-06-26 DOI: 10.1007/s40315-024-00551-6
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Abstract: Abstract Let f be an entire function and L(f) a linear differential polynomial in f with constant coefficients. Suppose that f, \(f'\) , and L(f) share a meromorphic function \(\alpha (z)\) that is a small function with respect to f. A characterization of the possibilities that may arise was recently obtained by Lahiri. However, one case leaves open many possibilities. We show that this case has more structure than might have been expected, and that a more detailed study of this case involves, among other things, Stirling numbers of the first and second kinds. We prove that the function \(\alpha \) must satisfy a linear homogeneous differential equation with specific coefficients involving only three free parameters, and then f can be obtained from each solution. Examples suggest that only rarely do single-valued solutions \(\alpha (z)\) exist, and even then they are not always small functions for f. PubDate: 2024-06-22 DOI: 10.1007/s40315-024-00552-5
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Abstract: Abstract In this paper, we shall prove the following result: Let f be a transcendental meromorphic function, and let P(z) be a polynomial with \(\deg P\ge 2\) . Then \([P(f(z))]^{(k)}\) takes every non-zero complex value infinitely often for \(k = 1, 2, 3, \ldots \) , by making use of, among other things, particularly an important result of Yamanoi (Proc Lond Math Soc 106:703–780, 2013). This improves the results due to Mues (Arch Math 32:55–67, 1979), Bergweiler and Eremenko (Rev Mat Iber 22:355–373, 1995), etc. Moreover, the corresponding normality criterion is also obtained. PubDate: 2024-06-11 DOI: 10.1007/s40315-024-00544-5
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Abstract: Abstract It is a classical theorem that if a function on the unit circle is integrable, then it is the nontangential limit of a holomorphic function on the open disc (subject to a certain growth condition) if and only if its Fourier coefficients for nonnegative integers are zero. In this article we generalize this result to higher complex dimensions by proving that for an integrable function on the unit sphere, it is a “boundary trace” of a holomorphic function on the open unit ball if and only if two particular families of integral equations are satisfied. To do this, we use the theory of Hardy spaces as well as the invariant Poisson and Cauchy integrals. PubDate: 2024-06-07 DOI: 10.1007/s40315-024-00549-0
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Abstract: Abstract A boundary integral equation method is presented for fast computation of the analytic capacities of compact sets in the complex plane. The method is based on using the Kerzman–Stein integral equation to compute the Szegő kernel and then the value of the derivative of the Ahlfors map at the point at infinity. The proposed method can be used for domains with smooth and piecewise smooth boundaries. When combined with conformal mappings, the method can be used for compact slit sets. Several numerical examples are presented to demonstrate the efficiency of the proposed method. We recover some known exact results and corroborate the conjectural subadditivity property of analytic capacity. PubDate: 2024-06-07 DOI: 10.1007/s40315-024-00547-2
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Abstract: Abstract Let \(0\le n\le 5\) and assume that \(L_n\) denotes the n-th Laguerre polynomial. We prove the following sharp inequality $$\begin{aligned} \int _{\mathbb {C}} \frac{ f^{(n)}(z) ^2e^{-\pi z ^2}}{\pi ^n n ! L_n(-\pi z ^2)}dxdy \le \Vert f\Vert ^2_{2,\pi }, \end{aligned}$$ for every holomorphic function f that belongs to the Fock space \(\mathcal {F}_\pi ^2\) . We conjecture that this is true for every integer \(n\geqslant 0\) . As an application we obtain the following sharp inequality $$\begin{aligned} \int _{\Omega } \frac{ f^{(n)}(z) ^2e^{-\pi z ^2}}{\pi ^n n ! L_n(-\pi z ^2)}dxdy \le \left( 1-e^{-(n+1) \Omega }\right) \Vert f\Vert ^2_{2,\pi }. \end{aligned}$$ Here \(\Omega \) is a domain in the complex plane and \( \Omega \) is its Lebesgue measure. The last inequality coincides with a recent result proved by Nicola and Tilli (Invent Math 230(1):1–30, 2022) for \(n=0\) . PubDate: 2024-06-01 DOI: 10.1007/s40315-023-00493-5
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Abstract: Abstract R. Küstner proved in his 2002 paper that the function $$\begin{aligned} w_{a,b,c}(z)= \frac{{}_2F_1(a +1,b;c;z)}{{}_2F_1(a,b;c;z)} \end{aligned}$$ maps the unit disk \( z <1\) onto a domain convex in the direction of the imaginary axis under a certain condition on the real parameters a, b, c. Here \({_2F_1}(a,b;c;z)\) stands for the Gaussian hypergeometric function. In this paper, we study the order of convexity of \(w_{a,b,c}\) . In particular, we partially solve the problem raised in the afore-mentioned paper by Küstner. PubDate: 2024-06-01 DOI: 10.1007/s40315-023-00489-1
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Abstract: Abstract In this work maximal-simultaneous approximation properties of the partial sums of Faber series in the Bergman space of analytic functions defined on bounded continuums of the complex plane are studied. The error of this approximation in dependence of the best approximation number and parameters of the considered canonical domains is estimated. PubDate: 2024-06-01 DOI: 10.1007/s40315-023-00496-2
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Abstract: Abstract Let X be a compact subset of the complex plane and let \(R^p(X)\) , \(2< p < \infty \) , denote the closure of the rational functions with poles off X in the \(L^p\) norm. In this paper we consider three conditions that show how the functions in \(R^p(X)\) can have a greater degree of smoothness at the boundary of X than might otherwise be expected. We will show that two of the conditions are equivalent and imply the third but the third does not imply the other two. PubDate: 2024-06-01 DOI: 10.1007/s40315-023-00503-6
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Abstract: Abstract We consider a two-dimensional equilibrium measure problem under the presence of quadratic potentials with a point charge and derive the explicit shape of the associated droplets. This particularly shows that the topology of the droplets reveals a phase transition: (i) in the post-critical case, the droplets are doubly connected domain; (ii) in the critical case, they contain two merging type singular boundary points; (iii) in the pre-critical case, they consist of two disconnected components. From the random matrix theory point of view, our results provide the limiting spectral distribution of the complex and symplectic elliptic Ginibre ensembles conditioned to have zero eigenvalues, which can also be interpreted as a non-Hermitian extension of the Marchenko–Pastur law. PubDate: 2024-06-01 DOI: 10.1007/s40315-023-00494-4
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Abstract: Abstract For a circle C contained in the unit disk, the necessary and sufficient condition for the existence of a triangle inscribed in the unit circle and circumscribed about C is known as Chapple’s formula. The geometric properties of Blaschke products of degree 3 given by Daepp et al. (Amer. Math. Monthly 109:785–794, 2002, https://doi.org/10.2307/3072367) and Frantz (Amer. Math. Monthly 111:779–790, 2004, https://doi.org/10.1080/00029890.2004.11920141) allow us to extend Chapple’s formula to the case of ellipses in the unit disk. The main aim of this paper is to provide a further extension of Chapple’s formula. Introducing a Blaschke-like map of a domain whose boundary is a conic, we extend their results to the case where the outer curve is an ellipse or a parabola. Moreover, we also give some geometrical properties for the Blaschke-like maps of degree d. PubDate: 2024-06-01 DOI: 10.1007/s40315-023-00499-z
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Abstract: Abstract The paper presents general criteria for the uniqueness of a non-constant meromorphic function having finitely many poles and a non-constant L-function in the Selberg class when they share two sets. Our results provide the best cardinalities ever obtained in the literature improving all the existing results Li et al. (Lith. Math. J. 58(2), 249–262 (2018)), Kundu and Banerjee (Rend. Circ. Mat. Palermo (2) 70(3), 1227–1244 (2021), Banerjee and Kundu (Lith. Math. J. 61(2), 161–179 (2021) with regard to the most general setting. Further, we have exhibited a number of examples throughout the paper showing the far reaching applications of our results. PubDate: 2024-06-01 DOI: 10.1007/s40315-023-00513-4
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Abstract: Abstract The stability of representations of univariate Lagrange interpolation polynomials in the complex plane is measured through a condition number. We study the growth of the condition number of the Newton formula for Lagrange interpolation. We prove that the condition number of Newton’s formula at the first n points of a Leja sequence for the closed unit disk \(\overline{{\mathbb {D}}}\) is bounded by \((n^3+2n-3)/3\) from above and by n/2 from below. We also point out that the condition number corresponding to any \(n+1\) distinct points on the unit circle is greater than \(n^c\) , where \(0<c<1\) is an absolute constant. PubDate: 2024-06-01 DOI: 10.1007/s40315-023-00497-1
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