on the capacity region $\mathcal{C}$ of the Y-channel. In order to find an upper bound on the sum capacity $C_g$, we have to solve $$\begin{aligned}
\max_{{\ensuremath{\mathbf{R}}}\in\overline{\mathcal{C}}} R_\Sigma,\end{aligned}$$ or otherwise, we can add any three bounds from Corollary \[CSG\] and Theorem \[BCBounds\] whose left hand side terms add to $R_\Sigma$. However, such an upper bound will depend on the relative value of $P_r$ compared to $P$. If we specialize these results to the Y-channel with $P=P_r$ we get a simpler representation for a sum-capacity upper bound. By combining the bounds in Corollary \[CSG\] and the bounds in Theorem \[BCBounds\], we obtain the following corollary.

\[Corollary:CutSetBounds\] If $P=P_r$, then the sum-capacity of the Y-channel is upper bounded by $\overline{C}_{\Sigma}$, i.e. $$\begin{aligned}
C_g\leq\overline{C}_{\Sigma}&\triangleq 2C(h_2^2P)+C( h_3^2P).\end{aligned}$$

By evaluating the bounds in corollary \[CSG\] for $P=P_r$, we have $$\begin{aligned}
\label{CSBnd1}
R_{12}+R_{13}&\leq C(\min\{h_1^2,h_2^2+h_3^2\}P),\\
\label{CSBnd2}
R_{21}+R_{23}&\leq C(\min\{h_2^2,h_1^2+h_3^2\}P),\\
\label{CSBnd3}
R_{31}+R_{32}&\leq C(\min\{h_3^2,h_1^2+h_2^2\}P).\end{aligned}$$ Moreover, from Theorem \[BCBounds\], we have $$\begin{aligned}
\label{BCBnd1}
R_{12}+R_{13}&\leq C(h_2^2P),\end{aligned}$$ if $P=P_r$, which is more binding than (\[CSBnd1\]) due to (\[Ordering\]). Adding (\[CSBnd2\]), (\[CSBnd3\]), and (\[BCBnd1\]) and using (\[Ordering\]) we obtain $$\begin{aligned}
R_\Sigma&\leq C(h_2^2P)+C(h_2^2P)+C(h_3^2P)\end{aligned}$$ and the statement of the corollary follows.

The upper bound in Corollary \[Corollary:CutSetBounds\] is independent of $h_1$ due to the assumption in (\[Ordering\]).

