On the complex oscillation for a class of homogeneous linear differential equations

Using a combined dominant condition, we obtain general results concerning t he complex oscillation for a class of homogeneous linear differential equat ions w((k)) + A(k-2)w((k-2)) + ... + A(1)w' + (A(0) + A)w = 0 with k greate r than or equal to 2, which has been investigated by many authors. In parti cular, we discover that there exists a unique case that possesses k linearl y independent zero-free solutions for these equations, and ae resolve an op en problem and simultaneously answer a question of Bank.