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Martin Chuaqui

Publications and source records attributed to Martin Chuaqui.

16 recordsLinked to original sources
arXiv2025

On the localization of the poles of the best Mobius approximations of f

We study the localization of the poles of the best Mobius approximations for locally univalent functions in the unit disk. Sharp geometric bounds for the pole function are established in terms of Pommerenke's linear invariant orders, refining classical criteria for convexity and concavity. The behavior of poles is further analyzed for starlike mappings, convex functions of order alpha, Janowski functions, and Robertson's class. For polygonal mappings, we describe the regions covered by the poles and obtain exact multiplicity results. We also derive new convexity conditions based on bounds of the Schwarzian derivative.

math.CV↗
arXiv2025

On Hardy spaces, univalent functions and the second coefficient

We consider normalized univalent functions with prescribed second Taylor coefficient $a_2$. For convex functions $f$ we study the Hardy spaces to which $f$ and $f'$ belong, refining in particular on a theorem of Eenigenburg and Keogh, and give a sharp asymptotic estimate and an explicit uniform bound for their coefficients. Relating the lower order of a convex function to the angle at infinity of its range we deduce that its range lies always in some sector of aperture $|a_2|π$. We give sharp smoothness conditions on the boundary for convex functions with prescribed second coefficient. We find the sharp Hardy space estimates for $f$ and $f'$ when $f$ belongs to other geometric subclasses, such as those of starlike, close-to-convex, convex in one direction, convex in the positive direction and typically real funtions. We extend a theorem of Lohwater, Piranian and Rudin, in which a univalent function whose derivative has radial limits almost nowhere is constructed, by showing that this pathological behavior can be obtained for any prescribed value of the second coefficient, in particular, manifesting itself arbitrarily close to the Koebe function.

math.CV↗
arXiv2025

The Ahlfors-Weill reflection on convex domains and Nehari quasidisks

The estimate $$\RR\{a_2f\}>-\frac12$$ derived for convex mappings in \cite{FMR}, is interpreted here in terms of the Ahlfors-Weill reflection to show that for such domains $\Om$, the mediatrix of the segment $[w, \mR_w]$ joining a point $w\in\Om$ and its reflection $\mR_w$ lies always outside $\Om$. In particular, the midpoint of the segment is also outside $\Om$. We determine the extremal cases when such a midpoint can lie of the boundary $\partial\Om$. The normalization $\fd=\frac{f}{1+a_2f}$ to a Möbius equivalent mapping with vanishing second coefficient leads to important distinctions between bounded an unbounded domains. We finally derive a geometric characterization of Nehari quasidisks in terms of the distance to the boundary of the Ahlfors-Weill reflection.

math.CV↗
arXiv2023

On the convolution of convex 2-gons

We study the convolution of functions of the form \[ f_α(z) := \dfrac{\left( \frac{1 + z}{1 - z} \right)^α- 1}{2 α}, \] which map the open unit disk of the complex plane onto polygons of 2 edges when $α\in(0,1)$. We extend results by Cima by studying limits of convolutions of finitely many $f_α$ and by considering the convolution of arbitrary unbounded convex mappings. The analysis for the latter is based on the notion of angle at infinity, which provides an estimate for the growth at infinity and determines whether the convolution is bounded or not. A generalization to an arbitrary number of factors shows that the convolution of $n$ randomly chosen unbounded convex mappings has a probability of $1/n!$ of remaining unbounded. We also extend Cima's analysis on the coefficients of the functions $f_α$ by providing precise asymptotic behavior for all $α$.

math.CV↗
arXiv2023

Best Möbius approximations of convex and concave mappings

We study the best Möbius approximations (BMA) to convex and concave conformal mappings of the disk, including the special case of mappings onto convex polygons. The crucial factor is the location of the poles of the BMAs. Finer details are possible in the case of polygons through special properties of Blaschke products and the prevertices of the mapping function.

math.CV↗
arXiv2023

Families of Homomorphic Mappings in the Polydisk

We study classes of locally biholomorphic mappings defined in the $¶$ that have bounded Schwarzian operator in the Bergman metric. We establish important properties of specific solutions of the associated system of differential equations and show a geometric connection between the order of the classes and a covering property. We show for modified and slightly larger classes that the order is Lipschitz continuous with respect to the bound on the Schwarzian, and use this to estimate the order of the original classes.

math.CV↗
arXiv2018

On convex mappings

We establish a new characterization for a conformal mapping of the unit disk $\mathbb{D}$ to be convex, and identify the mappings onto a half-plane or a parallel strip as extremals. We also show that, with these exceptions, the level sets of $λ$ of the Poincaré metric $λ|dw|$ of a convex domain are strictly convex.

math.CV↗
arXiv2016

Injectivity of minimal immersions and homeomorphic extensions to space

We study a recent general criterion for the injectivity of the conformal immersion of a Riemannian manifold into higher dimensional Euclidean space, and show how it gives rise to important conditions for Weierstrass-Ennerper lifts defined in the unit disk $\mathbb{D}$ endowed with a conformal metric. Among the corollaries, we obtain a Becker type condition and a sharp condition depending on the Gaussian curvature and the diameter for an immersed geodesically convex minimal disk in $\mathbb{R}^3$ to be embedded. Extremal configurations for the criteria are also determined, and can only occur on a catenoid. For non-extremal configurations, we establish fibrations of space by circles in domain and range that give a geometric analogue of the Ahlfors-Weill extension.

math.DG↗
arXiv2014

On Schwarz-Christoffel Mappings

We extend previous work on Schwarz-Chrsitoffel mappings, including the special cases when the image is a convex polygon or its complement. We center our analysis on the relationship between the pre-Schwrazian of such mappings and Blaschke products. For arbitrary Schwarz-Christoffel mappings, we resolve an open question in \cite{ChDO2} that relates the degrees of the associated Blaschke products with the number of convex and concave vertices of the polygon. In addition, we obtain a sharp sufficient condition in terms of the exterior angles for the injectivity of a mapping given by the Schwarz-Christoffel formula, and study the geometric interplay between the location of the zeros of the Blaschke products and the separation of the pre-vertices.

math.CV↗
arXiv2014

Affine and linear invariant families of harmonic mappings

We study the order of affine and linear invariant families of planar harmonic mappings in the unit disk and determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class $\mathcal{S}_H$ of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of $\mathcal{S}_H$ and the Schwarzian norm of the harmonic Koebe function.

math.CV↗
arXiv2014

Quasiconformal Extensions to Space of Weierstrass-Enneper Lifts

We derive a quasiconformal extension to 3-space of the Weierstrass-Enneper lifts of a class of harmonic mappings defined in the unit disk. The extension is based on fibrations of space by circles in domain and image that correspond to each other in a natural way. Convexity plays an essential role in the analysis. As a corollary we derive a sufficient condition for the underlying harmonic mapping to be univalent in the disk, with an explicit quasiconformal extension to the extended plane that generalizes the well known formula by Ahlfors-Weill.

math.CV↗
arXiv2012

The order of a linearly invariant familiy in C^n

We study the (trace) norm of a linearly invariant family in the ball in $\C$. By adapting an approach that in one variable yields optimal results, we are able to derive an upper bound for the norm of the family in terms of the Schwarzian norm and the dimension $n$.

math.CV↗
arXiv2010

Two-Point Distortion Theorems for Harmonic Mappings

In earlier work the authors have extended Nehari's well-known Schwarzian derivative criterion for univalence of analytic functions to a univalence criterion for canonical lifts of harmonic mappings to minimal surfaces. The present paper develops some quantitative versions of that result in the form of two-point distortion theorems. Along the way some distortion theorems for curves in ${\Bbb R}^n$ are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form.

math.CV↗
arXiv2010

Ahlfors-Weill Extensions for a Class of Minimal Surfaces

The Ahlfors-Weill extension of a conformal mapping of the disk is generalized to the lift of a harmonic mapping of the disk to a minimal surface, producing homeomorphic and quasiconformal extensions. The extension is obtained by a reflection across the boundary of the surface using a family of Euclidean circles orthogonal to the surface. This gives a geometric generalization of the Ahlfors-Weill formula and extends the minimal surface. Thus one obtains a homeomorphism of $\overline{\mathbb{C}}$ onto a toplological sphere in $\overline{\mathbb{R}^3} = \mathbb{R}^3 \cup \{\infty\}$ that is real-analytic off the boundary. The hypotheses involve bounds on a generalized Schwarzian derivative for harmonic mappings in term of the hyperbolic metric of the disk and the Gaussian curvature of the minimal surface. Hyperbolic convexity plays a crucial role.

math.CV↗
arXiv2007

Schwarzian Derivatives and Uniform Local Univalence

Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.

math.CV↗