  We introduce a new technique that allows us to make progress on two long
standing conjectures in transcendental dynamics: Baker's conjecture that a
transcendental entire function of order less than 1/2 has no unbounded Fatou
components, and Eremenko's conjecture that all the components of the escaping
set of an entire function are unbounded. We show that both conjectures hold for
many transcendental entire functions whose zeros all lie on the negative real
axis, in particular those of order less than 1/2. Our proofs use a classical
distortion theorem based on contraction of the hyperbolic metric, together with
new results which show that the images of certain curves must wind many times
round the origin.
