Browsing by Author "Chuaqui, Martin"
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- 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
- 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.
- 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 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.
- 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).