Generalized Schwarzian derivatives and higher order differential equations
Loading...
Date
2011
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
CAMBRIDGE UNIV PRESS
Abstract
It 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
s(f) = (f ''/f')' - 1/2(f ''/f')(2)
of f = h(1)/h(2), can be extended to the equation h((k)) + B(z) h = 0 where k >= 2. This generalization depends upon an appropriate definition of the generalized Schwarzian derivative S-k (f) of a function f which is induced by k - 1 ratios of linearly independent solutions of h((k)) + B(z) h = 0. The class R-k(Omega) of meromorphic functions f such that S-k(f) is analytic in a given domain Omega is also completely described. It is shown that if Omega is the unit disc D or the complex plane C, then the order of growth of f is an element of R-k(Omega) is precisely determined by the growth of S-k(f), and vice versa. Also the oscillation of solutions of h((k)) + B(z) h = 0, with the analytic coefficient B in D or C, in terms of the exponent of convergence of solutions is briefly discussed.
s(f) = (f ''/f')' - 1/2(f ''/f')(2)
of f = h(1)/h(2), can be extended to the equation h((k)) + B(z) h = 0 where k >= 2. This generalization depends upon an appropriate definition of the generalized Schwarzian derivative S-k (f) of a function f which is induced by k - 1 ratios of linearly independent solutions of h((k)) + B(z) h = 0. The class R-k(Omega) of meromorphic functions f such that S-k(f) is analytic in a given domain Omega is also completely described. It is shown that if Omega is the unit disc D or the complex plane C, then the order of growth of f is an element of R-k(Omega) is precisely determined by the growth of S-k(f), and vice versa. Also the oscillation of solutions of h((k)) + B(z) h = 0, with the analytic coefficient B in D or C, in terms of the exponent of convergence of solutions is briefly discussed.
Description
Keywords
UNIVALENT-FUNCTIONS, UNIT DISC, SPACES, COEFFICIENTS, GROWTH