  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 $\gamma'\leq(\log\rho-K_2)/c$.
