## 2 Notations and Preliminaries  
To make the paper self-contained, in this section, we provide some previous results that play a crucial role in our study. Throughout the present study, \(\mathbb{C}[z]\) denotes the ring of the univariate polynomials over \(\mathbb{C}\), the vector space \(\mathbb{R}^{n}\) is endowed with the standard coordinate-wise ordering \(\preceq\) defined by  
\[x\preceq y\quad\text{if and only if}\quad x_{i}\leq y_{i}\quad\text{for each}\ \ i=1,\ldots,n\]  
and the space \(\mathbb{C}^{n}\) is equipped with \(p\)-norm \(\|\cdot\|_{p}\) for some \(1\leq p\leq\infty\) and with the vector norm \(\|\cdot\|\) defined by  
\[\|x\|_{p}=\left(\sum_{i=1}^{n}|x_{i}|^{p}\right)^{1/p}\quad\text{and}\quad\| x\|=(|x_{1}|,\ldots,|x_{n}|).\]  
In addition, we define the functions \(d\colon\mathbb{C}^{n}\to\mathbb{R}^{n}\) and \(\delta\colon\mathbb{C}^{n}\to\mathbb{R}_{+}\) by  
\[d(x)=\left(d_{1}(x),\ldots,d_{n}(x)\right)\ \text{ with}\ \ d_{i}(x)=\min_{j\
eq i}|x_{i}-x_{j}|\ \text{ and}\ \ \delta(x)=\min_{i\
eq j}d_{i}(x) \tag{13}\]  
and, for two vectors \(x\in\mathbb{C}^{n}\) and \(y\in\mathbb{R}^{n}\), we use the denotation \(x/y\) for the vector in \(\mathbb{R}^{n}\) defined by  
\[\frac{x}{y}=\left(\frac{|x_{1}|}{y_{1}},\ldots,\frac{|x_{n}|}{y_{n}}\right)\]  
provided that \(y\) has only nonzero components. For a given number \(p\) (\(1\leq p\leq\infty\)), we define the number \(q\) by  
\[1\leq q\leq\infty\quad\text{with}\quad 1/p+1/q=1\]and, for a given \(n\in\mathbb{N}\), we use the denotations:  
\[a=(n-1)^{1/q},\quad b=2^{1/q}. \tag{14}\]  
Note that \(1\leq a\leq n-1\) and \(1\leq b\leq 2\). For a non-negative integer \(k\) and \(r\geq 1\), \(S_{k}(r)\) stands for the sum  
\[S_{k}(r)=\sum_{0\leq j<k}r^{j}.\]  
In addition, we assume that \(0^{0}\equiv 1\) and we denote by \(\mathbb{R}_{+}\) the set of non-negative numbers and, with \(J\), some interval in \(\mathbb{R}_{+}\) containing \(0\). Finally, a vector \(\xi\in\mathbb{C}^{n}\) is called _root-vector_ of \(f\) if  
\[f(z)=a_{0}\prod_{i=1}^{n}(z-\xi_{i})\quad\text{for all}\;\;z\in\mathbb{C}.\]  
**Definition 1** ([23; 24]). : _Let \(J\) be an interval on \(\mathbb{R}_{+}\) containing \(0\). A function \(\varphi\colon J\to\mathbb{R}_{+}\) is said to be quasi-homogeneous of exact degree \(m\geq 0\) if it satisfies the following two conditions:_  
* \(\varphi(\lambda t)\leq\lambda^{m}\varphi(t)\quad\text{for all}\;\;\lambda\in[0,1]\;\text{and}\;\;t\in J\)_;_
* \(\lim_{t\to 0^{+}}\frac{\varphi(t)}{t^{m}}\
eq 0\)_._  
The next definitions outline two important classes of iteration functions that will be used onwards. **Definition 2** ([24] (Definition 9)). : _A function \(F\colon D\subset\mathbb{C}^{n}\to\mathbb{C}^{n}\) is said to be an iteration function of first kind at a point \(\xi\in\mathcal{D}\) if there is a quasi-homogeneous function \(\phi\colon J\to\mathbb{R}_{+}\) of exact degree \(m\geq 0\) such that, for each vector \(x\in\mathbb{C}^{n}\) with \(E(x)\in J\), the following conditions hold:_  
\[x\in D\quad\text{and}\quad\left\|\,F(x)-\xi\,\right\|\preceq\phi(E(x))\, \left\|x-\xi\,\right\|, \tag{15}\]  
_where the function \(E\colon\mathbb{C}^{n}\to\mathbb{R}_{+}\) is defined by_  
\[E(x)=\left\|\frac{x-\xi}{d(\xi)}\right\|_{p}\quad(1\leq p\leq\infty) \tag{16}\]  
_with \(d\) defined by (13). The function \(\phi\) is said to be the control function of \(F\)._  
**Definition 3** ([24] (Definition 10)).