Eremenko's conjecture for functions with real zeros: the role of the minimum modulus

We show that for many families of transcendental entire functions $f$ the property that $m^n(r)\to\infty$ as $n\to \infty$, for some $r>0$, where $m(r)=\min\{|f(z)|:|z|=r\}$, implies that the escaping set $I(f)$ of $f$ has the structure of a spider's web. In particular, in this situation $I(f)$ is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.