Browsing by Author "Chuaqui, Martin"
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- ItemBest Möbius Approximations of Convex and Concave Mappings(2023) Chuaqui, Martin; Osgood, BradWe study the best Mobius 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.
- ItemFamilies of homomorphic mappings in the polydisk(2024) Chuaqui, Martin; Hernandez, RodrigoWe study classes of locally biholomorphic mappings defined in the polydisk P-n 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.
- ItemGeneralized Schwarzian derivatives and higher order differential equations(CAMBRIDGE UNIV PRESS, 2011) Chuaqui, Martin; Grohn, Janne; Rattya, JouniIt is shown that the well-known connection between the second order linear differential equation h '' B(z) h = 0, with a solution base {h(1), h(2)}, and the Schwarzian derivative
- ItemINTEGRAL CONDITIONS ON THE SCHWARZIAN FOR CURVES TO BE SIMPLE OR UNKNOTTED(2010) Chuaqui, MartinBy considering integral bounds on the Schwarzian derivative we extend previous results on sufficient conditions for curves in euclidean spaces to be simple or unknotted. The conditions are optimal.
- ItemMobius parametrizations of curves in R-n(BIRKHAUSER VERLAG AG, 2009) Chuaqui, MartinWe use Ahlfors' definition of Schwarzian derivative for curves in euclidean spaces to present new results about Mobius or projective parametrizations. The class of such parametrizations is invariant under compositions with Mobius transformations, and the resulting curves are simple. The analysis is based on the oscillatory behavior of the associated linear equation u '' + 1/4k(2)u = 0, where k = k(s) is the curvature as a function of arclength.
- ItemON AHLFORS' IMAGINARY SCHWARZIAN(2021) Chuaqui, MartinWe study geometric aspects of the imaginary Schwarzian S2f for curves in 3-space, as introduced by Ahlfors in [1]. We show that S2f points in the direction from the center of the osculating sphere to the point of contact with the curve. We also establish an important law of transformation of S2f under Mobius transformations. We finally study questions of existence and uniqueness up to Mobius transformations of curves with given real and imaginary Schwarzians. We show that curves with the same generic imaginary Schwarzian are equal provided they agree to second order at one point, while prescribing in addition the real Schwarzian becomes an overdetermined problem.
- ItemOn Ahlfors' Schwarzian derivative and knots(2007) Chuaqui, MartinWe extend Ahlfors' definition of the Schwarzian derivative for curves in euclidean space to include curves on arbitrary manifolds, and give applications to the classical spaces of constant curvature. We also derive in terms of the Schwarzian a sharp criterion for a closed curve in R-3 to be unknotted.
- ItemOn convolution, convex, and starlike mappings(2022) Chuaqui, Martin; Osgood, BradLet C and S* stand for the classes of convex and starlike mapping in D, and let <(co(C))over bar>, <(co(S*))over bar> denote the closures of the respective convex hulls. We derive characterizations for when the convolution of mappings in <(co(C))over bar> is convex, as well as when the convolution of mappings in <(co(S*))over bar> is starlike. Several characterizations in terms of convolution are given for convexity within <(co(C))over bar> and for starlikeness within <(co(S*))over bar>. We also obtain a correspondence via convolution between C and S*, as well as correspondences between the subclasses of convex and starlike mappings that have n-fold symmetry.
- ItemOn the convolution of convex 2-gons(2024) Chuaqui, Martin; Hernandez, Rodrigo; Llinares, Adrian; Mas, AlejandroWe study the convolution of functions of the form f alpha ( z) := /I 1+ z \ alpha - 1 1 -z , 2 alpha which map the open unit disk of the complex plane onto polygons of 2 edges when alpha is an element of (0 , 1). Inspired by a work of Cima, we study the limits of convolutions of finitely many f alpha and 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 provide the precise asymptotic behavior of the coefficients of the functions f alpha . (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
- ItemOSCILLATION OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS(CAMBRIDGE UNIV PRESS, 2009) Chuaqui, Martin; Duren, Peter; Osgood, Brad; Stowe, DennisIn this note we study the zeros of solutions of differential equations of the form u '' + pu = 0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.
- ItemPossible Intervals for T- and M-Orders of Solutions of Linear Differential Equations in the Unit Disc(HINDAWI PUBLISHING CORPORATION, 2011) Chuaqui, Martin; Grohn, Janne; Heittokangas, Janne; Rattya, JouniIn the case of the complex plane, it is known that there exists a finite set of rational numbers containing all possible growth orders of solutions of f((k)) + a(k-1)(z)f((k-1)) + ... + a(1)(z)f' + a(0)(z)f = 0 with polynomial coefficients. In the present paper, it is shown by an example that a unit disc counterpart of such finite set does not contain all possible T- and M-orders of solutions, with respect to Nevanlinna characteristic and maximum modulus, if the coefficients are analytic functions belonging either to weighted Bergman spaces or to weighted Hardy spaces. In contrast to a finite set, possible intervals for T- and M-orders are introduced to give detailed information about the growth of solutions. Finally, these findings yield sharp lower bounds for the sums of T- and M-orders of functions in the solution bases.
- ItemSchwarzian derivative criteria for valence of analytic and harmonic mappings(2007) Chuaqui, Martin; Duren, Peter; Osgood, BradFor analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass-Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally, certain classes of harmonic mappings are shown to have finite Schwarzian norm.
- ItemSCHWARZIAN DERIVATIVES OF CONVEX MAPPINGS(SUOMALAINEN TIEDEAKATEMIA, 2011) Chuaqui, Martin; Duren, Peter; Osgood, BradA simple proof is given for Nehari's theorem that an analytic function f which maps the unit disk onto a convex region has Schwarzian norm parallel to f parallel to <= 2. The inequality in sharper form leads to the conclusion that no convex mapping with parallel to f parallel to = 2 can map onto a quasidisk. In particular, every bounded convex mapping has Schwarzian norm parallel to f parallel to < 2. The analysis involves a structural formula for the pre-Schwarzian of a convex mapping, which is studied in further detail.
- ItemSCHWARZIAN NORMS AND TWO-POINT DISTORTION(2011) Chuaqui, Martin; Duren, Peter; Ma, William; Mejia, Diego; Minda, David; Osgood, BradAn analytic function f with Schwarzian norm parallel to gf parallel to <= 2(1 + delta(2)) is shown to satisfy a pair of two-point distortion conditions, one giving a lower bound and the other an upper bound for the deviation. Conversely, each of these conditions is found to imply that parallel to gf parallel to <= 2(1 + delta(2)). Analogues of the lower bound are also developed for curves in R-n and for canonical lifts of harmonic mappings to minimal surfaces.
- ItemTWO-POINT DISTORTION THEOREMS FOR HARMONIC MAPPINGS(2009) Chuaqui, Martin; Duren, Peter; Osgood, BradIn 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 R(n) are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form.
- ItemValence and oscillation of functions in the unit disk(SUOMALAINEN TIEDEAKATEMIA, 2008) Chuaqui, Martin; Stowe, DennisWe investigate the number of times that nontrivial solutions of equations u '' + p(z)u = 0 in the unit disk can vanish-or, equivalently, the number of times that solutions of S(f) = 2p(z) can attain their values-given a restriction vertical bar p(z)vertical bar < b(vertical bar z vertical bar). We establish a bound for that number when b satisfies a Nehari-type condition, identify perturbations of the condition that, allow the number to be infinite, and compare those results with their analogs for real equations phi '' + q(t)phi = 0 in (-1, 1).