In this work, we study the dynamics of Steffensen’s family of iterative root-finding methods for entire functions. This method is denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Lambda _{\beta ,f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mrow> <mi>β</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> for an entire function f. We first show that, for any entire function f excluding constants and linear polynomials, the Julia set of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Lambda _{\beta ,f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mrow> <mi>β</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is connected. It is also shown that Steffensen’s family of iterative root-finding methods does not satisfy the Scaling theorem for any polynomial. We prove that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$F(\Lambda _{\beta ,f})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow> <mi>β</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has an unbounded attracting domain for certain polynomials. It is shown that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F(\Lambda _{\beta ,f})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Λ</mi> <mrow> <mi>β</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has infinitely many attracting domains for any periodic transcendental entire function f having at least one zero. Finally, we give a class of meromorphic functions having simply connected wandering domains.

On the dynamics of Steffensen’s family of root-finding methods for entire function

期刊论文
发表日期：

2025年6月25日

查看全文

Nitai Mandal,
Subham Chatterjee,
Gorachand Chakraborty

摘要

In this work, we study the dynamics of Steffensen’s family of iterative root-finding methods for entire functions. This method is denoted by $\Lambda _{\beta ,f}$ $Λ_{β, f}$ for an entire function f. We first show that, for any entire function f excluding constants and linear polynomials, the Julia set of $\Lambda _{\beta ,f}$ $Λ_{β, f}$ is connected. It is also shown that Steffensen’s family of iterative root-finding methods does not satisfy the Scaling theorem for any polynomial. We prove that $F(\Lambda _{\beta ,f})$ $F (Λ_{β, f})$ has an unbounded attracting domain for certain polynomials. It is shown that $F(\Lambda _{\beta ,f})$ $F (Λ_{β, f})$ has infinitely many attracting domains for any periodic transcendental entire function f having at least one zero. Finally, we give a class of meromorphic functions having simply connected wandering domains.