Abstract: Abstract We extend representation formulas that generalize the similarity principle of solutions to the Vekua equation to certain classes of meta-analytic functions. Also, we solve a generalization of the higher-order Schwarz boundary value problem in the context of meta-analytic functions with boundary conditions that are boundary values in the sense of distributions. PubDate: 2024-06-12

Abstract: Abstract The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function \(V_{E}\) of a subset E of \({\mathbb {C}}^n\) , also called a pluricomplex Green function or global extremal function of E, equals the logarithm of the Siciak function \(\Phi _E\) if E is compact. The Siciak-Zakharyuta function is defined as the upper envelope of functions in the Lelong class that are negative on E, and the Siciak function is the upper envelope of m-th roots of the modulus of polynomials p in \(\mathcal {P}_m({\mathbb {C}}^n)\) of degree \(\le m\) such that \( p \le 1\) on E. We generalize the Siciak-Zakharyuta theorem to the case where the polynomial space \(\mathcal {P}_m({\mathbb {C}}^n)\) is replaced by \(\mathcal {P}_m^S({\mathbb {C}}^n)\) consisting of all polynomials with exponents restricted to sets mS, where S is a compact convex subset of \({\mathbb {R}}^n_+\) with \(0\in S\) . It states that if q is an admissible weight on a closed set E in \({\mathbb {C}}^n\) then \(V^S_{E,q}=\log \Phi ^S_{E,q}\) on \({\mathbb {C}}^{*n}\) if and only if the rational points in S form a dense subset of S. PubDate: 2024-06-05

Abstract: Abstract We introduce a class \({\mathfrak {R}}_{\alpha }^{q}\) of q-prestarlike functions of order \(\alpha \) by using q-difference operator. We obtain necessary and sufficient conditions involving convolution for functions to be in the class \({\mathfrak {R}}_{\alpha }^{q}.\) We prove that the well-known class of analytic prestarlike functions, \({\mathfrak {R}}_{\alpha },\) is properly contained in \({\mathfrak {R}}_{\alpha }^{q}.\) Apart from finding bounds on some initial coefficients of functions in the class \({\mathfrak {R}}_{\alpha }^{q}\) , we also investigate some convolution properties of functions in the class \({\mathfrak {R}}_{\alpha }^{q}.\) The results of the present manuscript essentially generalize some well-known results in the literature. PubDate: 2024-04-02

Abstract: Abstract The notion of supershift generalizes that one of superoscillation and expresses the fact that the sampling of a function in an interval allows to compute the values of the function outside the interval. In a previous paper, we discussed the case in which the sampling of the function is regular and we are considering supershift in a bounded set, while here we investigate how irregularity in the sampling may affect the answer to the question of whether there is any relation between supershift and real analyticity on the whole real line. We show that the restriction to \(\mathbb {R}\) of any entire function displays supershift, whereas the converse is, in general, not true. We conjecture that the converse is true as long as the sampling is regular, we discuss examples in support and we prove that the conjecture is indeed true for periodic functions. PubDate: 2024-04-01

Abstract: Abstract Let \(P(z)=\prod _{j=1}^{n}(z-z_j)\) be a polynomial of degree n with \( z_j \le k_j\le 1,1\le j\le n\) , it is known that $$\begin{aligned} \max _{ z =1} P^\prime (z) \ge \frac{n}{2}\left\{ 1+\frac{1}{1+\left( \frac{2}{n}\right) \sum _{j=1}^{n}\frac{k_j }{1-k_j}}\right\} \max _{ z =1} P(z) . \end{aligned}$$ In this paper, we shall extend this inequality to the generalized derivative of a polynomial without perturbing the condition on the zeros of underlying polynomial. We shall also present the polar derivative analogue of the result obtained for the generalized derivative of the polynomial. Our results include different relevant inequalities as special cases. PubDate: 2024-03-26

Abstract: Abstract Due to the invariance properties of cross-ratio, Möbius transformations are often used to map a set of points or other geometric object into a symmetric position to simplify a problem studied. However, when the points are mapped under a Möbius transformation, the distortion of the Euclidean geometry is rarely considered. Here, we identify several cases where the distortion caused by this symmetrization can be measured in terms of the Lipschitz constant of the Möbius transformation in the Euclidean or the chordal metric. PubDate: 2024-03-25

Abstract: Abstract A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group, in this case, the irreducible representation spaces of homogeneous harmonic polynomials. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with \(L^p\) boundary data, \(1 \le p \le \infty \) . We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy’s estimates, the mean-value property, Liouville’s Theorem, etc. PubDate: 2024-03-21

Abstract: Abstract Suppose V is a singular complex analytic curve inside \(\mathbb {C}^{2}\) . We investigate when a singular or non-singular complex analytic curve W inside \(\mathbb {C}^{2}\) with sufficiently small Hausdorff distance \(d_{H}(V, W)\) from V must intersect V. We obtain a sufficient condition on W which when satisfied gives an affirmative answer to our question. More precisely, we show the intersection is non-empty for any such W that admits at most one non-normal crossing type discriminant point associated with some proper projection. As an application, we prove a special case of the higher dimensional analog and also a holomorphic multifunction analog of a result by Lyubich and Peters (Geom. Funct. Anal. 24, 887–915 (2014)). PubDate: 2024-03-21

Abstract: Abstract We present a new complex analytic proof of the two classical formulas evaluating the sum of powers of consecutive integers which involve Stirling or Eulerian numbers. Our method generalizes that recently obtained by the second and third author for the sum of squares. PubDate: 2024-02-29

Abstract: Abstract We prove a topological decomposition of the space of meromorphic germs at zero in several variables with prescribed linear poles as a sum of spaces of holomorphic and polar germs. Evaluating the resulting holomorphic projection at zero gives rise to a continuous evaluator (at zero) on the space of meromorphic germs in several variables. Our constructions are carried out in the framework of Silva spaces and use an inner product on the underlying space of variables. They generalise to several variables, the topological direct decomposition of meromorphic germs at zero as sums of holomorphic and polar germs previously derived by the first and third author and provide a topological refinement of a known algebraic decomposition of such spaces previously derived by the second author and collaborators. PubDate: 2024-01-31

Abstract: Abstract Bernstein-type inequalities play a very important role in the theory of approximation. During last few decades, a number of operators have been identified, which preserve such type of inequalities between polynomials. In this paper, we consider a more general operator belonging to the family \({\mathcal {B}}_n\) and establish some inequalities preserved by it, which generalize or refine some of the results proved earlier. PubDate: 2024-01-30

Abstract: Abstract We study generalized quasiconformal mappings in the context of the inverse Poletsky inequality. We consider the local behavior and the boundary behavior of mappings with the inverse Poletsky inequality. In particular, we obtain logarithmic Hölder continuity for such classes of mappings. PubDate: 2024-01-27

Abstract: Abstract Local CR-diffeomorphisms between standard Hermitian quadrics of codimension 2 in \(\mathbb {C}^m\) are treated using a method based on the fundamental theorem of projective geometry. The cases when such diffeomorphisms must be projective are distinguished. PubDate: 2023-12-18

Abstract: Abstract We introduce a new distance on a domain \(\Omega \subset {{\mathbb {C}}}^n\) using the ‘minimizer’ functions on \({{\mathcal {A}}}^p(\Omega )\) . We discuss its invariance, completeness and other aspects related to it. PubDate: 2023-11-30

Abstract: Abstract In light of the Alexander transformation, the class of spirallike functions is significant. The characteristics of special functions also appear very frequently in Geometric function theory. In this paper, we find the radii of \(\gamma \) -spirallike and convex \(\gamma \) -spirallike of order \(\alpha \) of certain special functions. PubDate: 2023-11-27

Abstract: Abstract We consider mixed-norm Bergman spaces on homogeneous Siegel domains. In the literature, two different approaches have been considered and several results seem difficult to be compared. In this paper, we compare the results available in the literature and complete the existing ones in one of the two settings. The results we present are as follows: natural inclusions, density, completeness, reproducing properties, sampling, atomic decomposition, duality, continuity of Bergman projectors, boundary values, and transference. PubDate: 2023-10-31

Abstract: Abstract We consider a class of holomorphic skew product germs at super-saddle fixed points, which formally conjugate to their normal forms. We give necessary and sufficient conditions for the formal conjugacies to be holomorphic. PubDate: 2023-10-25

Abstract: Abstract In this paper, we introduce the notion of global \(L^q\) Gevrey vectors and investigate the regularity of such vectors in global and microglobal settings when \(q=2\) . We characterize the vectors in terms of the FBI transform and prove global and microglobal versions of the Kotake–Narasimhan Theorem. Our techniques are new because our results are written in terms of the FBI transform and not the Fourier transform. Additionally, the microglobal Kotake–Narasimhan Theorem provides a refinement of an earlier result by Hoepfner and Raich relating the microglobal wavefront sets of the ultradistributions u and Pu when P is a constant coefficient differential operator. PubDate: 2023-08-25

Abstract: Abstract We study the equivalence problem of classifying real-analytic second-order ordinary differential equations \(y_{xx}=J(x,y,y_{x})\) modulo fibre-preserving point transformations \(x\longmapsto \varphi (x)\) , \(y\longmapsto \psi (x,y)\) by using Moser’s method of normal forms. We first compute a basis of the Lie algebra \(\mathfrak {g}_{\{y_{xx}=0\}}\) of fibre-preserving symmetries of \(y_{xx}=0\) . In the formal theory of Moser’s method, this Lie algebra is used to give an explicit description of the set of normal forms \({{\mathscr {N}}}\) , and we show that the set is an ideal in the space of formal power series. We then show the existence of the normal forms by studying flows of suitable vector fields with appropriate corrections by the Cauchy–Kovalevskaya theorem. As an application, we show how normal forms can be used to prove that the identical vanishing of Hsu–Kamran primary invariants directly imply that the second-order differential equation is fibre-preserving point equivalent to \(y_{xx}=0\) . PubDate: 2023-08-14

Abstract: Abstract We prove a theorem which provides a sufficient condition for the non-existence of a complete Kähler–Einstein metric of negative scalar curvature of which holomorphic sectional curvature is negatively pinched: Let \(\Omega \) be a bounded weakly pseudoconvex domain in \(\mathbb {C}^n\) with a Kähler metric \(\omega \) whose holomorphic sectional curvature is negative near the topological boundary of \(\Omega \) (with respect to the relative topology of \(\mathbb {C}^n\) ) and \(\omega \) admits quasi-bounded geometry. Then \(\omega \) is uniformly equivalent to the Kobayashi–Royden metric and the following dichotomy holds: \(\omega \) is complete, and \(\omega \) is uniformly equivalent to the complete Kähler–Einstein metric with negative scalar curvature. \(\omega \) is incomplete, and there is no complete Kähler metric with negatively pinched holomorphic sectional curvature. Moreover, \(\Omega \) is Carathéodory incomplete. Our approach is based on the construction of a Kähler metric of negatively pinched holomorphic sectional curvature and applying the implication of equivalence of invariant metrics inspired by Wu-Yau. PubDate: 2023-06-07