## 3 Convergence Analysis
### Local Convergence of the First Kind  
Let \(f\in\mathbb{C}[z]\) be a polynomial of degree \(n\geq 2\) with \(n\) simple roots and let \(\xi\in\mathbb{C}^{n}\) be a root-vector of \(f\). In this section, we study the convergence of the iteration (10) regarding the function of initial conditions \(E\colon\mathbb{C}^{n}\to\mathbb{R}_{+}\) defined by (16). Let \(a\) and \(b\) be the numbers (14). For the needs of the main result, we define the real functions \(\gamma,\eta,l\colon[0,1/b)\to[1,\infty)\) by  
\[\gamma(t)=\left(1+\frac{a\,t}{(n-1)(1-bt)}\right)^{n-1},\quad\eta(t)=\frac{a \,t^{2}}{(1-t)(1-bt)}\quad\text{and}\quad\mu(t)=(\gamma(t)-1)^{2}+\gamma(t)^{ 2}\,\eta(t). \tag{27}\]  
Using these functions, we define \(\phi_{\alpha}\) by  
\[\phi_{\alpha}(t)=\left\{ \begin{array}{ll}\frac{|1-\alpha|\,\mu(t)+|\alpha|\, \gamma(t)\,\eta(t)}{|1-\alpha|-|\alpha|\,\gamma(t)\,(1+\eta(t))}&\text{for}\, \,\,Re(\alpha)<1/2,\\ \frac{|\alpha|\,\
u(t)^{2}+\eta(t)\,(1+|\alpha|\,\
u(t))}{(1-\eta(t))(1-| \alpha|\,\
u(t))}&\text{for}\,\,\,Re(\alpha)=1/2,\\ \frac{|1-\alpha|\,\mu(t)+|\alpha|\,\eta(t)\,(1-t)^{n-1}}{|\alpha|\,(1-\eta(t)) (1-t)^{n-1}-|1-\alpha|}&\text{for}\,\,\,Re(\alpha)>1/2,\end{array} \right. \tag{28}\]  
where \(\
u\) is defined by  
\[v(t)=\frac{a\,\gamma(t)\,t}{1-b\,t}. \tag{29}\]  
The following lemma shows that \(T_{\alpha}\) is an iteration function of first kind at a point \(\xi\). Lemma 5: _Let \(f\in\mathbb{C}[z]\) be a polynomial of degree \(n\geq 2\) that has only simple zeros in \(\mathbb{C}\), and let \(\xi\in\mathbb{C}^{n}\) be a root-vector of \(f\). Then, \(T_{\alpha}\colon D\subset\mathbb{C}^{n}\to\mathbb{C}^{n}\) defined by (11) is an iteration function of first kind at \(\xi\) with control function \(\phi_{\alpha}\) defined by (28)._  
Proof: First, let us define the functions \(g_{\alpha}\), \(h_{\alpha}\) and \(l_{\alpha}\) by  
\[g_{\alpha}(t) = |1-\alpha|-|\alpha|\,\gamma(t)\,(1+\eta(t)),\] \[h_{\alpha}(t) = (1-\eta(t))(1-|\alpha|\,v(t)),\] \[l_{\alpha}(t) = |\alpha|\,(1-\eta(t))(1-t)^{n-1}-|1-\alpha|. \tag{30}\]  
Now, let \(x\in\mathbb{C}^{n}\) be such that \(E(x)<1/b\) with \(g_{\alpha}(t)>0\) for \(Re(\alpha)<1/2\), \(h_{\alpha}(t)>0\) for \(Re(\alpha)=1/2\) or \(l_{\alpha}(t)>0\) for \(Re(\alpha)>1/2\). According to Definition 2 and Lemma 4, we ought to prove that  
\[x\in D\quad\text{and}\quad|\sigma_{i}|\leq\phi_{\alpha}(E(x))\quad\text{for each}\quad i=1,\dots,n, \tag{31}\]  
where the number \(\sigma_{i}\) is defined by (23). We divide the proof into two cases. **Case 1**. : _Let \(Re(\alpha)=1/2\). Note that, in this case, we have \(|\alpha|=|1-\alpha|\). Thus, from the first identity of (23), the triangle inequality and the following inequalities:_  
\[|B_{i}|\leq\frac{a\,E(x)^{2}}{(1-E(x))(1-bE(x))}=\eta(E(x))\quad\text{and} \quad|C_{i}|\leq\frac{a\gamma(E(x))E(x)}{1-bE(x)}=\
u(E(x)) \tag{32}\]  
_we obtain_  
\[|\sigma_{i}|\leq\frac{|1-\alpha|\,|C_{i}|^{2}+|1+\alpha C_{i}|\,|B_{i}|}{|1-B_ {i}|\,|1+\alpha C_{i}|}\leq\frac{|\alpha|\,\
u(E(x))^{2}+\eta(E(x))\,(1+| \alpha|\,\
u(E(x)))}{h_{\alpha}(E(x))}=\phi_{\alpha}(E(x))\]  
_which completes the proof of this case. Observe that the first inequality of (32) can be found in [24] (Lemma 4) whereas the second one follows from Lemma 1 and the known inequalities (see, e.g., [26] (Equation (30)))_  
\[|W_{j}|\leq\gamma(E(x))|x_{j}-\xi_{j}|\quad\text{and}\quad\sum_{j\
eq i}\frac {x_{j}-\xi_{j}}{x_{i}-x_{j}}\leq\frac{aE(x)}{1-bE(x)}.\]  
**Case 2**. : _Let \(Re(\alpha)\
eq 1/2\).