In this paper, we are concerned with the problem of locating the zeros of polynomials and regular functions with quaternionic coefficients when their real and imaginary parts are restricted. The extended Schwarz’s lemma, the maximum modulus theorem, and the structure of the zero sets defined in the newly constructed theory of regular functions and polynomials of a quaternionic variable are used to deduce the bounds for the zeros of these polynomials and regular functions. Our findings generalise certain recently established results about the zero distribution for this subclass of regular functions.

中文翻译：

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系数受限的四元多项式的零点界

在本文中，我们关注当实部和虚部受到限制时，多项式和具有四元数系数的正则函数的零点定位问题。利用扩展的施瓦茨引理、最大模定理以及新构造的正则函数和四元数变量多项式理论中定义的零集结构来推导这些多项式和正则函数的零点上界。我们的研究结果概括了最近确定的关于该正则函数子类的零分布的某些结果。