## 3 Proof of the Main Results  
Recently, [PERSON] [25] has provided general local convergence theorems about the _Picard iteration_  
\[x^{(k+1)}=T(x^{(k)}),\qquad k=0,1,2,\ldots, \tag{18}\]  
where \(T\) is an iteration function. We start this section by recalling some crucial definitions and results from [5; 25] that are needed to prove of our main results. **Definition 1** ([25]). : _A function \(f\colon J\to\mathbb{R}_{+}\) is said to be quasi-homogeneous of exact degree \(m\geq 0\) on an interval \(J\subset\mathbb{R}_{+}\) containing \(0\) if it satisfies the following two conditions:_  
(i) \(f(\lambda t)\leq\lambda^{m}f(t)\) _for all_ \(\lambda\in[0,1]\) _and_ \(t\in J\)_;_  
(ii) \(\lim_{t\to 0^{+}}f(t)/t^{m}\
eq 0\)_._  
**Definition 2** ([25], Definition 10). : _A function \(F\colon D\subset\mathbb{C}^{n}\to\mathbb{C}^{n}\) is said to be an iteration function of second kind at a point \(\xi\in\mathbb{C}^{n}\) if there is a nonzero quasi-homogeneous function \(\beta\colon J\to\mathbb{R}_{+}\) such that for each vector \(x\in\mathcal{D}\) with \(E(x)\in J\), the following conditions hold:_  
\[x\in D\quad\text{and}\quad\|\,F(x)-\xi\,\|\preceq\beta(E(x))\,\|x-\xi\,\|, \tag{19}\]  
_where the function \(E\colon\mathcal{D}\to\mathbb{R}_{+}\) is defined by (10). The function \(\beta\) is said to be a control function of \(F\)._  
**Lemma 1** ([25], Example 1). : _Let \(n\in\mathbb{N}\) and \(f\) be a quasi-homogeneous function of exact degree \(r>0\) on an interval \(J\); then, the function_  
\[g(t)=(1+f(t))^{n}-1\]  
_is also a quasi-homogeneous function of exact degree \(r\) on \(J\)._  
**Lemma 2** ([5], Proposition 5.5). : _If \(u\in\mathbb{C}^{n}\) and \(1\leq p\leq\infty\), then_  
\[\left|\prod_{j=1}^{n}(1+u_{j})\right|\leq\left(1+\frac{\|u\|_{p}}{n^{1/p}} \right)^{n}\quad\text{and}\quad\left|\prod_{j=1}^{n}(1+u_{j})-1\right|\leq \left(1+\frac{\|u\|_{p}}{n^{1/p}}\right)^{n}-1.\]  
To prove Theorem 1, we shall apply the following general local convergence theorem of [PERSON] [25]. **Theorem 3** ([25], Theorem 4). : _Let \(T\colon D\subset\mathbb{C}^{n}\to\mathbb{C}^{n}\) be an iteration function of second kind at a point \(\xi\in\mathbb{C}^{n}\) with a nonzero control function \(\beta\colon J\to\mathbb{R}_{+}\) of exact degree \(m\geq 0\) and let \(x^{(0)}\in\mathbb{C}^{n}\) be an initial approximation with distinct components such that_  
\[E(x^{(0)})\in J\quad\text{and}\quad\Psi(E(x^{(0)}))\geq 0, \tag{20}\]_where \(\Psi\colon J\to\mathbb{R}\) is defined by_  
\[\Psi(t)=1-bt-\beta(t)(1+bt), \tag{21}\]  
_with \(b=2^{1/q}\). Then, \(\xi\) is a fixed point of \(T\) with pairwise distinct components and Picard iteration (18) is well defined and converges to \(\xi\) with \(Q\)-order \(r=m+1\) and with error estimates for all \(k\geq 0\):_  
\[\|x^{(k)}-\xi\|\preceq\theta^{k}\lambda^{S_{k}(r)}\,\|x^{(0)}-\xi\|\quad\text {and}\quad\|x^{(k+1)}-\xi\|\preceq\theta\lambda^{r^{k}}\,\|x^{(k)}-\xi\| \tag{22}\]  
_where \(\lambda=\phi(E(x^{(0)}))\), \(\theta=\psi(E(x^{(0)}))\) and the functions \(\psi\) and \(\phi\) are defined by_  
\[\psi(t)=1-bt(1+\beta(t))\quad\text{and}\quad\phi(t)=\beta(t)\,/\psi(t).\]  
_Also, the following a posteriori error estimate holds for all \(k\geq 0\):_  
\[\|x^{(k+1)}-\xi\,\|_{p}\leq c_{k}\,\|x^{(k)}-\xi\,\|_{p}^{r},\]  
_where the sequence \((c_{k})_{k=0}^{\infty}\) is defined by \(c_{k}=\sigma(E(x^{(k)}))/\delta(x^{(k)})^{m}\) and \(\sigma:J\to\mathbb{R}_{+}\) is defined by \(\sigma(0)=0\) and \(\sigma(t)=\beta(t)\,/t^{m}\) for \(t>0\)._  
The aim of the next lemma is to provide a sufficient condition for the modified Weierstrass iteration function (4) to be an iteration function of second kind at \(\xi\in\mathbb{C}^{n}\) with control function of first degree. **Lemma 3**. : _Let \(f\in\mathbb{C}[z]\) be a polynomial of degree \(n\geq 2\) with only simple roots and \(\xi\in\mathbb{C}^{n}\) be a root vector of \(f\).