+ 2 m \sum_{k=0}^{N-1} \gamma_k^{r+1} (C_f^2 + \lambda_k^2 C_h^2) 
     + 2 m^2 C_f \sum_{k=0}^{N-1} \gamma_k^{r+1} \left( C_f + \lambda_k C_h \right)  . 
    \end{aligned}$$ Rearranging the terms we have $$\begin{aligned}
    &2 \sum_{k=0}^{N-1} \gamma_k^r \left( f(x_k) - f^*\right)  \leq  2 m \sum_{k=0}^{N-1} \gamma_k^{r+1} (C_f^2 + \lambda_k^2 C_h^2) \\& 
    + 2 m^2 C_f \sum_{k=0}^{N-1} \gamma_k^{r+1} \left( C_f + \lambda_k C_h \right) + 4 M_h \sum_{k=0}^{N-1} \gamma_k^r \lambda_k + 4M^2\gamma_{N-1}^{r-1} . 
    \end{aligned}$$ Now we divide both sides by $2\sum_{k=0}^{N-1} \gamma_k^r$ and use the definition of $\psi_{k,N-1}$ in , $$\begin{aligned}
    &\sum_{k=0}^{N-1} \psi_{k,N-1} \left( f(x_k) - f^*\right) \\& \leq   \left(\sum_{k=0}^{N-1} \gamma_k^r\right)^{-1} \left( m \sum_{k=0}^{N-1} \gamma_k^{r+1} (C_f^2 + \lambda_k^2 C_h^2) \right. \cr\\& \left. 
    +m^2 C_f \sum_{k=0}^{N-1} \gamma_k^{r+1} \left( C_f + \lambda_k C_h \right) + 2 M_h  \sum_{k=0}^{N-1} \gamma_k^r \lambda_k + 2M^2\gamma_{N-1}^{r-1} \right). 
    \end{aligned}$$ We know $\sum_{k=0}^{N-1} \psi_{k,N-1}=1$ also $f(\bar x_N) \leq \sum_{k=0}^{N-1} \psi_{k,N-1}f(x_k)$ because of convexity of $f$. So, we obtain the desired result.

The following lemma, will be used to find a convergence rate statement in Theorem \[thm rate\].

\[Lemma 9, page 418 in [@Yousefian17]\]\[lemma:ineqHarmonic\] For any scalar $\alpha\neq -1$ and integers $\ell$ and $N$ where $0\leq \ell \leq N-1$, we have $$\begin{aligned}
    \frac{N^{\alpha+1}-(\ell+1)^{\alpha+1}}{\alpha+1}&\leq \sum_{k=\ell}^{N-1}(k+1)^\alpha \\& \leq (\ell+1)^\alpha+\frac{(N+1)^{\alpha+1}-(\ell+1)^{\alpha+1}}{\alpha+1}.
    \end{aligned}$$

In the following theorem[[,]{}]{} we present a 