Abstract
It is known that, if $f$ is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set $J(f)$ are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions $f_K$, $K \in \N$, such that the box and packing dimensions of $J(f_K)$ are equal to two, even though as $K \to \infty$ the Hausdorff dimension of $J(f_K)$ tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function.
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