It follows that: $$\begin{aligned}
 \sum_{m=0}^M Q_1^m &\hspace{-0.1in} =& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m}\max_{k\in\mathcal{U}_l^m}\left\{\Gamma_m / H_m^k\right\} \nonumber \\
 &\hspace{-0.1in} \le& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m} \overline{G}_m \nonumber \\
 &\hspace{-0.1in} \le& \hspace{-0.1in} N_0 \sum_{m=0}^M \overline{G}_m \sum_{l=1}^L (1+\Gamma_m)^{l-1} \nonumber \\
 &\hspace{-0.1in} =& \hspace{-0.1in} N_0 \sum_{m=0}^M \overline{G}_m \left[ (1+\Gamma_m)^L-1 \right] / \Gamma_m. \label{eq:UBound}\end{aligned}$$

In (\[eq:UBound\]), the first inequality is from the definition of $\overline{G}_m$. The second inequality is from the definition of $c_{l+1}^m$. Specifically, $c_1^m=0$; when $\mathcal{U}_l^m \neq \emptyset$, we have $c_{l}^m=c_{l-1}^m+1$; when $\mathcal{U}_l^m = \emptyset$, we have $c_{l}^m=c_{l-1}^m$. It follows that $c_l^m\le l-1$. Therefore, (\[eq:UBound\]) is an upper bound on the objective function (\[eq:ObjFun2\]).

Furthermore, by defining $\overline{G} = \max_{m\in\{0,\cdots,M\}} \left\{ \overline{G}_m \right\}$, and $\overline{\Gamma} = \max_{m\in\{0,\cdots,M\}} \left\{ \Gamma_m \right\}$, we can get a looser upper bound from \[eq:UBound\] as $$\sum_{m=0}^M Q_1^m \leq N_0 \overline{G} (M+1)\left[ (1+\overline{\Gamma})^L-1\right]/\overline{\Gamma}.$$

Next, we derive a lower bound for (\[eq:ObjFun2\]). Define $$\left\{ \begin{array}{l}
  \underline{G}^l = \min_{m\in\{0,\cdots,M\}} \max_{k\in\mathcal{U}_l^m} \left\{ \Gamma_m/ H_m^k \right\} \\
  \underline{\Gamma} = \min_{m\in\{0,\cdots,M\}} \left\{ \Gamma_m \right\}. 
          \end{array} \right.$$ We have that $$\begin{aligned}
 \sum_{m=0}^M Q_1^m &\hspace{-0.1in} =& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m} \max_{k\in\mathcal{U}_l^m} \left\{ \Gamma_m / H_m^k \right\} \nonumber \\
 &\hspace{-0.1in} \ge& \hspace{-0.1in} N_0 \sum_{m=0}^M \sum_{l=1}^L (1+\Gamma_m)^{c_l^m} \underline{G}^l \nonumber \\
 &\hspace{-0.1in} \ge& \hspace{-0.1in} N_0 \sum_{l=1}^L \underline{G}^l \sum_{m=0}^M 