## 2 Preliminaries  
The best-known semi-local convergence result for [PERSON]'s method under a Holder condition on the first derivative of the operator involved, when the technique of recurrence relations is used to prove it, is that given by [PERSON] in [4], which is established under the following conditions:  
1. There exists \(\Gamma_{0}=[F^{\prime}(x_{0})]^{-1}\in\mathcal{L}(Y,X)\), for some \(x_{0}\in\Omega\), with \(\|\Gamma_{0}\|\leq\beta\) and \(\|\Gamma_{0}F(x_{0})\|\leq\eta\), where \(\mathcal{L}(Y,X)\) is the set of bounded linear operators from \(Y\) to \(X\).
2. There exist two constants \(K\geq 0\) and \(p\in(0,1]\), such that \(\|F^{\prime}(x)-F^{\prime}(y)\|\leq K\|x-y\|^{p}\) for \(x,y\in\Omega\).
3. If \(\xi(p)\) is the unique zero of the auxiliary function: \[\phi(x;p)=(1+p)^{p}(1-x)^{1+p}-x^{p},\quad p\in(0,1],\] (2) in the interval \(\left(0,\frac{1}{2}\right]\), it is satisfied that \(\mathpzc{h}=K\beta\eta^{p}\leq\xi(p)\) and \(B(x_{0},R)\subset\Omega\), where \(R=\frac{(1+p)(1-\delta)}{(1+p)-(2+p)\delta}\eta\).  
**Theorem 1**.: _(Theorem 2.1 of [4]) Let \(F:\Omega\subseteq X\longrightarrow Y\) be a continuously-differentiable operator defined on a nonempty open convex domain \(\Omega\) of a Banach space \(X\) with values in a Banach space \(Y\). Suppose that Conditions (A1)(A3) are satisfied. Then, [PERSON]'s sequence, given by (1), converges to a solution \(x^{*}\) of the equation \(F(x)=0\), starting at \(x_{0}\), and \(x_{n},x^{*}\in\overline{B(x_{0},R)}\), for all \(n=0,1,2,\ldots\)_  
Note that Condition (A1), required for the initial approximation \(x_{0}\), defines the parameters \(\beta\) and \(\eta\), and Condition (A2), required for the operator \(F\), defines the parameter \(K\) fixed for all point of \(\Omega\). Observe that, every point \(z\in\Omega\), such that the operator \([F^{\prime}(z)]^{-1}\) exists with \(\|[F^{\prime}(z)]^{-1}\|\leq\beta\) and \(\|[F^{\prime}(z)]^{-1}F(z)\|\leq\eta\), has associated the pair \((K,\beta\eta^{p})\) of the \(xy\)-plane, where \(x=K\) and \(y=\beta\eta^{p}\). In addition, fixed \(p\in(0,1]\), if we consider the set:  
\[\mathcal{D}_{\mathsf{T}}=\left\{(x,y)\in\mathbb{R}^{2}:xy\leq\xi(p)\right\},\]  
we can observe that every point \(z\), such that its associated pair \((K,\beta\eta^{p})\) belongs to \(\mathcal{D}_{\mathsf{T}}\), can be chosen as the starting point for [PERSON]'s method, so that the method converges to a solution \(x^{*}\) of the equation \(F(x)=0\) when it starts at \(z\). The set \(\mathcal{D}_{\mathsf{T}}\) is then called the domain of parameters associated with Theorem 1 and can be drawn by choosing \(x=K\), \(y=\beta\eta^{p}\) and coloring the region of the \(xy\)-plane whose points satisfy the condition \(\mathpzc{h}\leq\xi(p)\) (namely, \(xy\leq\xi(p)\)) of the Theorem 1. In Figure 1, we see the domain of parameters \(\mathcal{D}_{\mathsf{T}}\) (blue region).  
In relation to the above, we can think that the larger the size of the domain of parameters is, the more possibilities we have for choosing good starting points for [PERSON]'s method. As a consequence, we are interested in \(\mathcal{D}_{\mathsf{T}}\) being as big as possible, since this fact allows us to find a greater number of good starting points for [PERSON]'s method.