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<p>In this paper, two distinct approaches are developed to establish Cauchy integral formulas and Schwarz-type lemmas for solutions to quaternion polynomial Dirac equations. First, by constructing a new fundamental solution, the first kind of Cauchy integral formulas are obtained. Then the corresponding Schwarz-type lemmas are derived by utilizing this Cauchy integral formula. Subsequently, the second kind of Cauchy integral formulas are established by exploiting the connection between <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\ker (D-\lambda )^{k}\)</EquationSource> </InlineEquation> and <i>k</i>-monogenic functions. Finally, the second kind of Schwarz-type lemmas are obtained.</p>

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Cauchy integral formulas and Schwarz-type lemmas for solutions to quaternion polynomial Dirac equations

  • 期刊论文

  • 发表日期：

    2026年1月15日

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  • Shuoxing He,
  • Chunxue Duan,
  • Yonghong Xie

摘要

In this paper, two distinct approaches are developed to establish Cauchy integral formulas and Schwarz-type lemmas for solutions to quaternion polynomial Dirac equations. First, by constructing a new fundamental solution, the first kind of Cauchy integral formulas are obtained. Then the corresponding Schwarz-type lemmas are derived by utilizing this Cauchy integral formula. Subsequently, the second kind of Cauchy integral formulas are established by exploiting the connection between \(\ker (D-\lambda )^{k}\) and k-monogenic functions. Finally, the second kind of Schwarz-type lemmas are obtained.