## 4 Accessibility of [PERSON]'s Method
In Figure 1, we see that \(\mathcal{D}_{\mathrm{T}}\subset\mathcal{D}_{\mathrm{C}}^{\varepsilon}\) for \(p=\frac{1}{10},\frac{1}{5},\frac{2}{5},\frac{4}{5}\), so that we can guess that the smaller the quantity \(\varepsilon=\frac{K_{0}}{K}\in[0,1]\) is, the larger the domain of parameters is: orange for \(\varepsilon=0.1\), green for \(\varepsilon=0.2\), red for \(\varepsilon=0.4\) and yellow for \(\varepsilon=0.8\). Note that, if \(\varepsilon\to 1\), the domain of parameters associated with Corollary 4, \(\mathcal{D}_{\mathrm{C}}^{\varepsilon}\), tends to be that obtained by Theorem 1 (blue region). As a consequence,  
\[\mathcal{D}_{\mathrm{T}}=\mathcal{D}_{\mathrm{C}}^{1}=\mathcal{D}_{\mathrm{C} }^{\varepsilon_{j}}\subset\mathcal{D}_{\mathrm{C}}^{\varepsilon_{j-1}} \subset\cdots\subset\mathcal{D}_{\mathrm{C}}^{\varepsilon_{0}}\qquad\text{ for}\qquad\varepsilon_{0}<\cdots<\varepsilon_{j-1}<\varepsilon_{j}=1.\]  
On the other hand, in Figure 2, we observe the relationship between the domains of parameters associated with Theorem 1 (gray region) and Corollary 4 (magenta region) from the variability of \(\varepsilon\) and for four different values of \(p\): \(p=\frac{1}{10},\frac{1}{5},\frac{2}{5},\frac{4}{5}\). As we can see, the domain associated with Corollary 4 is always larger, for all \(\varepsilon\in[0,1]\), than that associated with Theorem 1. Figure 1: Domains of parameters of [PERSON]’s method associated with Theorem 1 (blue region) and Corollary 4 (orange for \(\varepsilon=0.1\), green for \(\varepsilon=0.2\), red for \(\varepsilon=0.4\) and yellow for \(\varepsilon=0.8\)) for \(p=\frac{1}{10},\frac{1}{5},\frac{2}{5},\frac{4}{5}\). In addition, we prove analytically in the following what we have just seen graphically. First, we prove that \(\mathcal{D}_{\mathsf{T}}\subset\mathcal{D}_{\mathsf{C}}^{\varepsilon}\), for each \(p\in(0,1]\) and \(\varepsilon\in[0,1]\). For this, if \((K,\beta\eta^{p})\in\mathcal{D}_{\mathsf{T}}\), then \(\mathpzc{h}\leq\xi(p)\leq\frac{1}{2}\leq\frac{1}{2\varepsilon}\leq\frac{1+p}{ \varepsilon(2+p)}\), since \(p\in(0,1]\) and \(\varepsilon\in[0,1]\). Moreover, \(\mathpzc{h}\leq\xi(p)\leq\xi(p)\frac{(1+p)^{p}(1-\varepsilon\mathpzc{h})^{1+p}} {\varepsilon^{p}\mathpzc{h}^{p}}\), since \(\phi(\varepsilon\mathpzc{h};p)\geq 0\) when \(\mathpzc{h}\leq\xi(p)\) and \(\varepsilon\mathpzc{h}\leq\xi(p)\). Therefore, \((K,\beta\eta^{p})\in\mathcal{D}_{\mathsf{C}}^{\varepsilon}\) for each \(p\in(0,1]\) and \(\varepsilon\in[0,1]\). Finally, we see that \(\mathcal{D}_{\mathsf{C}}^{\varepsilon_{2}}\subset\mathcal{D}_{\mathsf{C}}^{ \varepsilon_{1}}\) if \(\varepsilon_{1}<\varepsilon_{2}\) with \(\varepsilon_{1},\varepsilon_{2}\in[0,1]\). Indeed, from \(\psi(\varepsilon)=\frac{1+p}{(2+p)\varepsilon}\) and \(\varphi(\varepsilon)=\xi(p)\frac{(1+p)^{p}(1-\varepsilon\mathpzc{h})^{1+p}} {\varepsilon^{p}\mathpzc{h}^{p}}\), we have \(\psi^{\prime}(\varepsilon)=\frac{-(1+p)^{p}\xi(p)(1-\varepsilon\mathpzc{h})^ {p}(p+\varepsilon\mathpzc{h})}{\varepsilon^{1+p}\mathpzc{h}^{p}}\leq 0\), since \(1-\varepsilon\mathpzc{h}\geq\frac{1}{2+p}\geq 0\), so that \(\psi(\varepsilon_{2})\leq\psi(\varepsilon_{1})\) and \(\varphi(\varepsilon_{2})\leq\varphi(\varepsilon_{1})\), and therefore \(\mathcal{D}_{\mathsf{C}}^{\varepsilon_{2}}\subset\mathcal{D}_{\mathsf{C}}^{ \varepsilon_{1}}\). To that end, we have proven the improvement obtained for the domain of parameters of [PERSON]'s method with the help of the conditions of type (3) that we have just shown in \[\phi(b_{1};p)>0. \tag{18}\]0