R_{jl}+R_{kl}&\leq C\left(\min\left\{(|h_j|+|h_k|)^2P,h_l^2P_r\right\}\right),\end{aligned}$$ for all distinct $j,k,l\in\{1,2,3\}$.

See Appendix \[CSGProof\].

In the following theorem, we give other bounds on the achievable rates in the Y-channel based on a degraded broadcast channel bound.

\[BCBounds\] The achievable rates in the Y-channel must satisfy $$\begin{aligned}
R_{12}+R_{13}&\leq C(h_2^2P_r),\\
R_{21}+R_{23}&\leq C(h_1^2P_r),\\
R_{31}+R_{32}&\leq C(h_1^2P_r).\end{aligned}$$

Let us give the relay all the messages as side information, i.e. the relay knows ${\ensuremath{\mathbf{m}}}$ apriori. And let us also give $(m_{31},m_{32})$ and $(m_{21},m_{23})$ as side information to receivers 2 and 3 respectively (see Figure \[GA\_BC\]). Now receivers 2 and 3 share the knowledge of $m_{21}$, $m_{23}$, $m_{31}$ and $m_{32}$ which are also known at the relay. The relay knows $m_{12}$ and $m_{13}$ which should be delivered to receivers 2 and 3 respectively. The resulting setup is a degraded broadcast channel (BC) whose sum-capacity is [@CoverThomas] $$\begin{aligned}
R_{12}+R_{13}\leq C(\max\{h_2^2,h_3^2\}P_r).\end{aligned}$$ Similarly, we can obtain bounds on $R_{21}+R_{23}$ and $R_{31}+R_{32}$. Using (\[Ordering\]), we obtain the statement of the theorem.

![Genie aided Y-Channel as a degraded broadcast channel.[]{data-label="GA_BC"}](Y_Channel_ga_bc.eps){width="0.8\columnwidth"}

The bounds in Corollary \[CSG\] and Theorem \[BCBounds\], in addition to $R_{jk}\geq0$ and the single user bounds $$\begin{aligned}
R_{jk}\leq \min\{C(h_j^2P),C(h_k^2P_r)\}\end{aligned}$$ for all distinct $j,k\in\{1,2,3\}$ form a region $\overline{\mathcal{C}}$ in the 6-dimensional space which is an outer bound 