noted that $\underline{C}^{II}$ can be larger than $\underline{C}^{III}$ in some cases, e.g. if $h_3=h_2$.

\[t\][$\mathsf{SNR}$(dB)]{} \[b\][Sum Rate (Bits/channel use)]{} \[l\] \[l\] \[l\] \[l\] \[l\] \[l\] ![A plot of the upper and lower bounds for a Y-channel with $P=P_r$, $h_1=1$, $h_2=0.8$, and $h_3=0.7$.[]{data-label="sum_rate_asymmetric"}](sum_rate_asymmetric_g.eps "fig:"){width="\columnwidth"}

Bounding the Gap between the Upper and Lower Bounds {#GapCalculation:General}
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The functional decode and forward scheme achieves the DoF of the Y-channel. This can be seen from the pre-log in the lower bound in Theorem \[LowerBound:FDF\] and the upper bounds in Theorem \[SRUBG\]. Now we bound the gap between the upper and lower bounds. We consider two kinds of gaps, additive gap and multiplicative gap.

We bound the multiplicative gap $\Gamma_m$ first. That is, we bound the ratio of the upper bound to the lower bound. For this purpose, we use the bounds $\overline{C}_\Sigma$ and $\underline{C}^I$. Notice that we can always write $$\begin{aligned}
C_g&\geq\underline{C}^{I}\geq C\left(h_2^2P\right).\end{aligned}$$ Therefore $$\begin{aligned}
\Gamma_{m}&=\frac{\overline{C}_\Sigma}{\underline{C}^I}\\
&\leq\frac{3C(h_2^2P)}{C(h_2^2P)}\\
&\leq 3.\end{aligned}$$

Now we calculate the additive gap, which we split into two cases: $h_2^2P\leq1/2$ and $h_2^2P>1/2$.

Case $h_2^2P\leq1/2$
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In this case, we call the gap $\Gamma_{a1}$. Consider the lower bound $\underline{C}^I$ and the upper bound $\overline{C}_\Sigma$. These bounds can be used to obtain the following. 