where ,, = Im, y(t) = ym(t) and m 1, , d. This was proved by Perron (cf. [8], pp. 158-159) with the lim replaced by lim sup, but his method can be used to supply (iv) as it stands, cf. [7]. In view of known results ([2], [3]) for the case when (i) is equivalent to a second order differential equation, it is natural to ask under what conditions, weaker than (iii), will the assertion concerning (iv) remain true. In this direction, it will be shown that (iii) can be relaxed to