relay channel. By choosing the strongest two users to communicate all the time, i.e. users 1 and 2, the following rates can be achieved [@NamChungLee] $$\begin{aligned}
R_{12}=R_{21}\leq\min\left\{\left[C\left(h_2^2P-\frac{1}{2}\right)\right]^+,C(h_2^2P_r)\right\}\end{aligned}$$ Thus, we can bound the sum-capacity as follows.

\[LowerBound:FDF\_2User\] The sum-capacity of the Y-channel satisfies $C_g\geq\underline{C}^{III}$ where $$\begin{aligned}
\underline{C}^{III}&=2\min\left\{\left[C\left(h_2^2P-\frac{1}{2}\right)\right]^+,C\left(h_2^2P_r\right)\right\}.\end{aligned}$$

Figure \[sum\_rate\_asymmetric\] shows a plot of the obtained upper and lower bounds for the case $P=P_r$ versus the signal to noise power ratio $\mathsf{SNR}$. Namely, the plotted bounds are: the upper bound obtained with the cut-set approach $\overline{C}_{\Sigma}$, the upper bound obtained with the genie aided approach $\overline{C}_{\Sigma g}$, the complete decode-and-forward lower bound $\underline{C}^I$, the functional decode-and-forward lower bound $\underline{C}^{II}$, and the functional decode-and-forward lower bound with two active users $\underline{C}^{III}$, for a Y-channel with $h_1=1$, $h_2=0.8$, $h_3=0.7$. It can be seen that $\overline{C}_{\Sigma g}$ is tighter than $\overline{C}_{\Sigma}$ at moderate to high $\mathsf{SNR}$. It can also be seen that the gap between $\overline{C}_{\Sigma g}$ and $\underline{C}^{II}$, $\underline{C}^{III}$ becomes constant as $\mathsf{SNR}$ increases. In the following section, we characterize this constant gap. Notice that the lower bound $\underline{C}^{III}$ is simpler than $\underline{C}^{II}$. For this reason, we will use $\underline{C}^{III}$ to characterize that gap between the upper and lower bounds. However, it must be 