## 1 Introduction  
In this study, we are concerned with the problem of approximating a locally-unique solution \(x^{*}\) of equation:  
\[F(x)=0 \tag{1}\]  
where \(F\) is a differentiable function defined on a convex subset \(\mathbb{D}\) of \(\mathbb{S}\) with values in \(\mathbb{S}\), where \(\mathbb{S}\) is \(\mathbb{R}\) or \(\mathbb{C}\). Many problems from applied sciences, including engineering, can be solved by means of finding the solutions of equations in a form like Equation (1) using mathematical modeling [2; 3; 4; 5; 6; 7]. Except in special cases, the solutions of these equations can be found in closed form. This is the main reason why the most commonly-used solution methods are usually iterative. The convergence analysis of iterative methods is usually divided into two categories: semi-local and local convergence analysis. The semi-local convergence matter is, based on the information around an initial point, to give criteria ensuring the convergence of iteration procedures. A very important problem in the study of iterative procedures is the radius of convergence. In general, the radius of convergence is small. Therefore, it is important to enlarge the radius of convergence. Another important problem is to find more precise error estimates on the distances \(\|x_{n}-x^{*}\|\). The most popular method for approximating a simple solution \(x^{*}\) of Equation (1) is undoubtedly [PERSON]'s method, which is given by:  
\[x_{n+1}=x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n}),\quad\text{ for each }n=0,1,2\ldots. \tag{2}\]  
provided that \(F^{\prime}\) does not vanish in \(\mathbb{D}\)[2, 13]. To obtain a higher order of convergence, many methods have been proposed [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41]. We study the local convergence of the three-step method defined for each \(n=0,1,2,\ldots\) by:  
\[ \begin{array}{lcl}y_{n}&=&x_{n}-F^{\prime}(x_{n})^{-1}F(x_{n}),\\ z_{n}&=&y_{n}-\frac{F(x_{n})+\beta F(y_{n})}{F(x_{n})+(\beta-2)F(y_{n})}F^{ \prime}(x_{n})^{-1}F(x_{n}),\\ x_{n+1}&=&x_{n}-\frac{F(z_{n})}{A_{n}},\end{array}  \tag{3}\]  
where \(x_{0}\) is an initial point, \(\beta\in\mathbb{S}\) and:  
\[ \begin{array}{l}A_{n}=2[x_{n},\;z_{n};\;F]-2[x_{n},\;y_{n};\;F]+[z_{n},\;y_{ n};\;F]+(y_{n}-z_{n})[y_{n},\;x_{n},\;x_{n};\;F],\\ [x_{n},\;y_{n};\;F]=\frac{F(x_{n})-F(y_{n})}{x_{n}-y_{n}},\text{and}\\ [y_{n},\;x_{n},\;x_{n};\;F]=\frac{[x_{n},\;y_{n};\;F]-F^{\prime}(x_{n})}{y_{n} -x_{n}}.\end{array} \]  
The eighth order of convergence for Method (3) was established in [1], when \(\beta\in\mathbb{S}\), using Taylor expansions and hypotheses reaching up to the eighth derivative of \(F\), although only the first derivatives and the divided difference appear in these methods. This method is also an optimal in the sense of Traub with efficiency index \(8^{\frac{1}{4}}\)[4]. The advantages of Method (3) over other competing methods were also shown in [1]. However, the hypotheses of higher order derivatives limit the applicability of these methods. As a motivational example, define function \(F\) on \(\mathbb{X}=\mathbb{Y}=\mathbb{R}\), \(D=[-\frac{5}{2},\frac{1}{2}]\) by:  
\[F(x)=\left\{ \begin{array}{ll}x^{3}lnx^{2}+x^{5}-x^{4},&x\
eq 0\\ 0,&x=0\end{array} .\right.\]  
Then, we have that:  
\[ \begin{array}{l}F^{\prime}(x)=3x^{2}lnx^{2}+5x^{4}-4x^{3}+2x^{2},\\ F^{\prime\prime}(x)=12 xlnx^{2}+20x^{3}-12x^{2}+10x\end{array} \]  
and:  
\[F^{\prime\prime\prime}(x)=12 lnx^{2}+60x^{2}-12x+22.\]Then, obviously, function \(F^{\prime\prime\prime}(x)\) is unbounded on \(\mathbb{D}\). Hence, the results in [1], cannot apply to show the convergence of Method (3) or its special cases requiring hypotheses on the third derivative of function \(F\) or higher.