Estimation of Bounds for the Zeros of Polynomials and Regular Functions of a Quaternionic Variable

The estimation of zeros of a polynomial with quaternionic coefficients has been done by many mathematicians in the recent past using various approaches. In this paper, we estimate the upper bounds for the zeros of polynomials and derive zero-free regions of some special regular functions of a quaternionic variable with restricted coefficients using the extended Schwarz’s lemma and the zero sets of a regular product. The obtained results for this subclass of polynomials and regular functions produce generalizations of a number of results known in the literature on this subject.