Much is known about the connection between the growth and decay of subharmonic functions. The results indicate that there is a general principle: {\em asubharmonic function cannot decay ‘too fast’ relative to its growth}.Three theorems are proved which, together with work previously published elsewhere, give a fairly complete account of how this principle works out for a subharmonic function having extremal decay along a ray.

1991 Mathematics Subject Classification: 30D20, 31A05.