On the complex oscillation of some linear differential equations

We treat the linear differential equation (*) f((k)) + A(z)f = 0, where k greater than or equal to 2 is an integer and A(z) is a transcendental entire function of order cr(A). It is shown that any non-trivial solution of the equation (*) satisfies lambda(f) greater than or equal to sigma(A), where lambda(f) is the exponent of convergence of the zero-sequence of f, under the condition <K(N)over bar>(r, 1/A) less than or equal to T(r, A), r is not an element of E for a K > 2k and an exceptional set E of finite linear measure. The second order equation f '' + (e(P1(z)) + e(P2(z)) + Q(z))f = 0, where P-1(z), P-2(z) are non-constant polynomials and Q(z) is an entire function, is also studied. (C) 1997 Academic Press.