Generalized Schwarzian derivatives and higher order differential equations

It is shown that the well-known connection between the second order linear differential equation h″ + B(z)h = 0, with a solution base {h1, h2}, and the Schwarzian derivative \begin{equation} S_f = {\left( \frac{f''}{f'} \right)}' - \frac{1}{2} \, {\left( \frac{f''}{f'} \right)}^2 \end{equation} of f = h1/h2, 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 Sk(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 Sk(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 Sk(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.