in Theorem \[SRUBG\]. Before we state this theorem, we need the following lemmas.

\[FromRelay\] The achievable rates in the Y-channel must satisfy $$\begin{aligned}
R_{kj}+R_{lj}+R_{kl}&\leq C(h_j^2P_r+h_l^2P_r)\end{aligned}$$ for all distinct $j,k,l\in\{1,2,3\}$.

We use a genie aided approach to bound the sum of three rates, e.g. $R_{21}+R_{31}+R_{23}$, by giving $m_{32}$ and $(Y_1^n,m_{21},m_{12},m_{13})$ as additional information to receivers 1 and 3 respectively. Details are given in Appendix \[ProofFromRelay\].

\[GUB\] The achievable rates in the Y-channel must satisfy $$\begin{aligned}
R_{kj}+R_{lj}+R_{kl}&\leq C((|h_k|+|h_l|)^2P)\end{aligned}$$ for all distinct $j,k,l\in\{1,2,3\}$.

We use a genie aided approach to bound the sum of three rates such as $R_{21}+R_{31}+R_{23}$ by giving $(Y_r^n,m_{32})$ and $(Y_r^n,m_{21},m_{12},m_{13})$ as additional information to receiver 1 and 3 respectively. See Appendix \[GeneralProof\] for more details.

As a result of Lemmas \[FromRelay\] and \[GUB\], we obtain $$\begin{aligned}
&R_{kj}+R_{lj}+R_{kl}\nonumber\\
\label{3BoundG}
&\hspace{0.5cm}\leq \min\left\{C(h_j^2P_r+h_l^2P_r),C((|h_k|+|h_l|)^2P)\right\}.\end{aligned}$$ Now if the Y-channel has $P=P_r$, we obtain the following sum-capacity upper bound.

\[SRUBG\] The sum-capacity of the Y-channel with $P=P_r$ is upper bounded by $\overline{C}_{\Sigma g}$, i.e. $$\begin{aligned}
C_g\leq\overline{C}_{\Sigma g}&=C(h_2^2P+h_3^2P)\nonumber\\
&\quad+C(\min\{h_1^2P+h_3^2P,(|h_2|+|h_3|)^2P\}).\end{aligned}$$

By evaluating (\[3BoundG\]) for $(j,k,l)=(2,1,3)$, and for $(j,k,l)=(1,3,2)$ and adding the two obtained bounds, we obtain the desired result.

As we can see, the bound in Theorem \[SRUBG\] has a pre-log equal to 1. The slope of 