  In transcendental dynamics significant progress has been made by studying
points whose iterates escape to infinity at least as fast as iterates of the
maximum modulus. Here we take the novel approach of studying points whose
iterates escape at least as fast as iterates of the {\it minimum} modulus, and
obtain new results related to Eremenko's conjecture and Baker's conjecture, and
the rate of escape in Baker domains. To do this we prove a result of wider
interest concerning the existence of points that escape to infinity under the
iteration of a positive continuous function.
