Oscillation results of certain higher order linear differential equations with periodic coefficients in the complex plane

Baesch (Results in Math. 29, 1996, 42-55) has given a characterization of homogeneous linear differential equations with certain analytic periodic coefficients which admits a solution with finite exponent of convergence. However, her method seems too general and in most cases too complicated for applications. We give, in this paper, a direct approach to the problem and obtain several such characterizations which do not seem to follow from those of Baesch. In particular, an explicit, necessary and sufficient condition to the problem is given for certain third order equations. The results again do not seem to follow from those of Baesch. Our method is based on that of Y. M. Chiang, I. Laine, and S. Wang (Complex Variables, 34 (1997), 25-34) which in turn depends on basic representations of solutions given by Bank, Laine, and Langley. (C) 1997 Academic Press