  Eremenko and Lyubich proved that an entire function whose set of singular
values is bounded is expanding at points where its image has large modulus.
These expansion properties have been at the centre of the subsequent study of
this class of functions, now called the Eremenko-Lyubich class. We improve the
estimate of Eremenko and Lyubich, and show that the new estimate is
asymptotically optimal. As a corollary, we obtain an elementary proof that
functions in the Eremenko-Lyubich class have lower order at least $1/2$.
