On the Hausdorff dimension of the set of non-escaping points in the Julia set

For an entire function $f$ and a non-zero complex number $\lambda$, let $f_{\lambda }(z)=f(\lambda z)$. We give conditions on $f$ which imply that the Hausdorff dimension of the set of non-escaping points in the Julia set of $f_{\lambda }$ tends to $1$ as $\mathrm{\mid }\lambda \mathrm{\mid }\to 0$. In fact, we give an upper bound for this dimension in terms of $\lambda$. This generalizes earlier results concerned with the case that $f(z)=\exp z$.