  For a transcendental entire function $f$, the property that there exists
$r>0$ such that $m^n(r)\to\infty$ as $n\to\infty$, where $m(r)=\min
\{|f(z)|:|z|=r\}$, is related to conjectures of Eremenko and of Baker, for both
of which order $1/2$ minimal type is a significant rate of growth. We show that
this property holds for functions of order $1/2$ minimal type if the maximum
modulus of $f$ has sufficiently regular growth and we give examples to show the
sharpness of our results by using a recent generalisation of Kjellberg's method
of constructing entire functions of small growth, which allows rather precise
control of $m(r)$.
