- x^* \right \|^2  - \gamma_{0}^{r-1} \|x_1 - x^*\|^2 + 4M^2\left( \gamma_{0}^{r-1}-\gamma_{N-1}^{r-1}\right)   \\& \leq 2 \sum_{k=1}^{N-1} \gamma_k^r \left( f^* - f(x_k)\right)  \label{ineq4 INC}+ 2 m \sum_{k=1}^{N-1} \gamma_k^{r+1} (C_f^2 + \lambda_k^2 C_h^2) 
    \\ & + 2 m^2 C_f \sum_{k=1}^{N-1} \gamma_k^{r+1} \left( C_f + \lambda_k C_h \right) + 4 M_h \sum_{k=1}^{N-1} \gamma_k^r \lambda_k .
    \end{aligned}$$ Removing non-negative terms from the left-hand side of the preceding inequality, we have $$\begin{aligned}
    &- \gamma_{0}^{r-1} \|x_1 - x^*\|^2  - 4M^2\gamma_{N-1}^{r-1}  \leq 2 \sum_{k=1}^{N-1} \gamma_k^r \left( f^* - f(x_k)\right) \nonumber \\ & + 2 m \sum_{k=1}^{N-1} \gamma_k^{r+1} (C_f^2 + \lambda_k^2 C_h^2) 
    + 2 m^2 C_f \sum_{k=1}^{N-1} \gamma_k^{r+1} \left( C_f + \lambda_k C_h \right) \nonumber \\& + 4 M_h \sum_{k=1}^{N-1} \gamma_k^r \lambda_k . 
    \end{aligned}$$ From for $k=0$, we obtain $$\begin{aligned}
    \left \|x_1 - x^* \right \|^2  & \leq 2\gamma_0 \left( f^* - f(x_0)\right) + 2 m \gamma_0^2 (C_f^2 + \lambda_0^2 C_h^2)  \\
    & + 2 m^2 C_f \gamma_0^2 \left( C_f + \lambda_0 C_h \right) + 4\gamma_0 \lambda_0 M_h + 4M^2. 
    \end{aligned}$$ By multiplying both sides of the preceding inequality by $\gamma_{0}^{r-1}$ and summing it with the relation , we obtain $$\begin{aligned}
    &- 4M^2\gamma_{N-1}^{r-1}  \leq 2 \sum_{k=0}^{N-1} \gamma_k^r \left( f^* - f(x_k)\right)  + 4 M_h \sum_{k=0}^{N-1} \gamma_k^r \lambda_k \\& 