The function $g$ can be written as $g=\frac{\sum_{m\in M}\alpha_{m}\chi^{m}}{\sum_{m\in M}\beta_{m}\chi^{m}}$. Then

	$\displaystyle\psi_{g}(u)$	$\displaystyle=\frac{1}{\lambda_{K}}\log|g(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))|$
		$\displaystyle=\frac{1}{\lambda_{K}}\log\Bigl|\sum_{m\in M}\alpha_{m}\chi^{m}(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))\Bigr|-\frac{1}{\lambda_{K}}\log\Bigl|\sum_{m\in M}\beta_{m}\chi^{m}(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))\Bigr|$
		$\displaystyle=\max_{m\in M}\Bigl(\frac{\log|\alpha_{m}|}{\lambda_{K}}-\left<m,u\right>\Bigr)-\max_{m\in M}\Bigl(\frac{\log|\beta_{m}|}{\lambda_{K}}-\left<m,u\right>\Bigr)$
		$\displaystyle=\max_{m\in M}(-{\operatorname{ord}}(\alpha_{m})-\left<m,u\right>)-\max_{m\in M}(-{\operatorname{ord}}(\beta_{m})-\left<m,u\right>)$
		$\displaystyle=\min_{m\in M}(\left<m,u\right>+{\operatorname{ord}}(\beta_{m}))-\min_{m\in M}(\left<m,u\right>+{\operatorname{ord}}(\alpha_{m})).$

Thus, it is the difference of two H-lattice concave functions. ∎