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 $m^n(r)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$, for some $r>0$, where $m(r)=\min \{\mathrm{\mid }f(z)\mathrm{\mid }:\mathrm{\mid }z\mathrm{\mid }=r\}$, 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.