  We consider the family of entire maps given by
$f_{\ell,c}(z)=\ell+c-(\ell-1)\log c-e^z$, where $c\in D(\ell,1)$ and
$\ell\in\mathbb N$, $\ell\geq2$. By using the property of $f_{\ell,c}$ to be
dynamically projected to an infinite cylinder $\mathbb C/2\pi I\mathbb Z$,
where the thermodynamic formalism tools are well-defined, we prove as a main
result on this work, the real-analyticity of the map $c\mapsto
HD(\mathcal{J}_r(f_{\ell,c}))$, here $\mathcal{J}_r(f_{\ell,c})$ is the radial
Julia set.
