Hausdorff measure of escaping and Julia sets for bounded type functions of finite order

We show that the escaping sets and the Julia sets of bounded type transcendental entire functions of order $\rho$ become 'smaller' as $\rho\to\infty$. More precisely, their Hausdorff measures are infinite with respect to the gauge function $h_\gamma(t)=t^2g(1/t)^\gamma$, where $g$ is the inverse of a linearizer of some exponential map and $\gamma\geq(\log\rho(f)+K_1)/c$, but for $\rho$ large enough, there exists a function $f_\rho$ of bounded type with order $\rho$ such that the Hausdorff measures of the escaping set and the Julia set of $f_\rho$ with respect to $h_{\gamma'}$ are zero whenever $\gamma'\leq(\log\rho-K_2)/c$.