arXiv subjects

Alexandre Eremenko

Publications and source records attributed to Alexandre Eremenko.

At least 19 recordsLinked to original sources

arXiv2026

When a meromorphic function that omits three values is of bounded type

Suppose that a function $F$ is meromorphic in the domain $\mathbb H(-m) = \{ z : \mathrm{Im}\, z > -m(\mathrm{Re}\, z) \}$, where $m$ is an even, positive, and continuous function that does not increase on $\mathbb R_{\ge 0}$, and suppose that $F$ omits there three distinct values. Then $F$ is of bounded type in the upper half-plane (i.e., is represented there as a quotient of two bounded analytic functions), provided that the logarithmic integral of the function $m$ is convergent. On the other hand, if the logarithmic integral of $m$ diverges, there exists a function $F$ meromorphic in $\mathbb H(-m)$, that omits there three distinct values, and which is of unbounded type in the upper half-plane. This result is motivated by a century old question originating with Rolf Nevanlinna.

math.CV↗

arXiv2025

Julia directions for holomorphic curves

A theorem of Picard's type is proved for entire holomorphic mappings into complex projective varieties. This theorem has local character in the sense that the existence of Julia directions can be proved under a natural additional assumption. An example is given which shows that Borel's theorem on holomorphic curves omitting hyperplanes does not have such a local counterpart.

math.CV↗

arXiv2024

Carath\'eodory convergence and the conformal type problem

We study Carath\'eodory convergence for open, simply connected surfaces spread over the sphere and, in particular, provide examples demonstrating that in the Speiser class the conformal type can change when two singular values collide.

math.CV↗

arXiv2024

On the Shapes of Rational Lemniscates

A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert's lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge's theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps.

math.CV↗

arXiv2024

The work of Walter Bergweiler in value distribution of meromorphic functions

This is a colloquium talk in CAU, Kiel delivered on June 7, 2024 on the occasion of Walter Bergweiler's retirement. Walter's work on meromorphic functions consists of two parts: generalizations of Picard's theorem to differential polynomials, and the applications of the rescaling principle known as the Bloch Principle. Since the talk was aimed at the general audience, a brief introduction to Nevanlinna theory is included.

math.CV↗

arXiv2024

Iossif Ostrovskii's work on entire functions

The theory of entire functions and its applications were at the center of Ostrovskii's research interests throughout his entire career. He made lasting contributions to several aspects of this theory, and many of his works had a significant influence on subsequent research. In this note, we describe some of this work.

math.CV↗

arXiv2023

On Bloch's "Principle of topological continuity''

We discuss to what extent certain results about totally ramified values of entire and meromorphic functions remain valid if one relaxes the hypothesis that some value is totally ramified by assuming only that all islands over some Jordan domain are multiple. In particular, we prove a result suggested by Bloch which says that an entire function of order less than $1$ has a simple island over at least one of two given Jordan domains with disjoint closures.

math.CV↗

arXiv2022

Zeros and coefficients

Two theorems on the asymptotic distribution of zeros of sequences of analytic functions are proved. First one relates the asymptotic behavior of zeros to the asymptotic behavior of coefficients. Second theorem establishes a relation between the asymptotic behaviors of zeros of a function and zeros of derivative.

math.CV↗

arXiv2022

On conformal metrics of constant positive curvature in the plane

We prove three theorems about solutions of $\Delta u + e^{2u} = 0$ in the plane. The first two describe explicitly all concave and quasiconcave solutions. The third theorem says that the diameter of the plane with respect to the metric with line element $e^{u}|dz|$ is at least $4\pi/3$, except for two explicitly described families of solutions u.

math.AP↗

arXiv2022

Rigidity results on Liouville equation

We give a complete classification of solutions bounded from above of the Liouville equation $$-\Delta u=e^{2u}\quad\mbox{in}\quad {\mathbf{R}}^2.$$ More generally, solutions in the class $$N:=\{ u:\limsup_{z\to\infty} u(z)/\log|z|:=k(u)<\infty\}$$ are described. As a consequence, we obtain five rigidity results. First, $k(u)$ can take only a discrete set of values: either $k=-2$, or $2k$ is a non-negative integer. Second, $u\to-\infty$ as $z\to\infty$, if and only if $u$ is radial about some point. Third, if $u$ is symmetric with respect to $x$ and $y$ axes and $u_x<0,\; u_y<0$ in the first quadrant then $u$ is radially symmetric. Fourth, if $u$ is concave and bounded from above, then $u$ is one-dimensional. Fifth, if $u$ is bounded from above, and the diameter of ${\mathbf{R}}^2$ with the metric $e^{2u}\delta$ is $\pi$, where $\delta$ is the Euclidean metric, then $u$ is either radial about a point or one-dimensional. In addition, we extend the concavity rigidity result on Liouville equation in higher dimensions.

math.AP↗

arXiv2022

Second order linear differential equations with a basis of solutions having only real zeros

Let $A$ be a transcendental entire function of finite order. We show that if the differential equation $w''+Aw=0$ has two linearly independent solutions with only real zeros, then the order of $A$ must be an odd integer or one half of an odd integer. Moreover, $A$ has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.

math.CV↗

arXiv2021

Geometric theory of meromorphic functions

This is a survey of results on the following problem. Consider a simply connected Riemann surface spread over the Riemann sphere. How are the properties of the uniformizing function of this surface related to the geometric properties of the surface? Based on the lecture in U. Michigan in May 2006.

math.CV↗

arXiv2021

Ahlfors' contribution to the theory of meromorphic functions

This is an expanded version of one of the Lectures in memory of Lars Ahlfors in Haifa in 1996. Some mistakes are corrected and references added. It contains a survey of his work on meromorphic functions and related topics written in 1929-1941.

math.CV↗

arXiv2021

Singularities of inverse functions

A survey of general results on the singularities of inverses to meromorphic functions is given, with applications to holomorphic dynamics. This is a lecture delivered at the workshop "The role of complex analysis in complex dynamics" in Edinburgh on May 22 2013, with corrected mistakes and updated references.

math.CV↗

arXiv2021

Stability in the Marcinkiewicz theorem

Ostrovskii's generalization of the Marcinkiewicz theorem implies that if an entire characteristic functions of a probability distribution satisfies $\log^+\log|f(z)|=o(|z|),\; z\to\infty,$ and is zero-free then the distribution is normal. We show that under the same growth condition, absence of zeros in a wide vertical strip implies that the distribution is close to a normal one. This generalizes and simplifies a recent result of Michelen and Sahasrabudhe.

math.PR↗

arXiv2021

Metrics of constant positive curvature with conic singularities. A survey

We consider conformal metrics of constant curvature 1 on a Riemann surface, with finitely many prescribed conic singularities and prescribed angles at these singularities. Especially interesting case which was studied by C. L. Chai, C. S Lin and C. L. Wang is described in some detail, with simplified proofs.

math.DG↗

arXiv2020

Canonical embeddings of pairs of arcs

We show that for given four points in the Riemann sphere and a given isotopy class of two disjoint arcs connecting these points in two pairs, there exists a unique configuration with the property that each arc is a hyperbolic geodesic segment in the complement of the other arc.

math.CV↗

arXiv2020

Meromorphic functions with three radially distributed values

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.

math.CV↗