2011/06/20 by MARTIN CHUAQUI, JANNE GRÖHN, JOUNI RÄTTYÄ
paper · doi:10.1017/s0305004111000296
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 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 k (Ω) of meromorphic functions f such that S k ( f ) is analytic in a given domain Ω is also completely described. It is shown that if Ω is the unit disc or the complex plane , then the order of growth of f ∈ k (Ω) 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 or , in terms of the exponent of convergence of solutions is briefly discussed.