Some properties of two analytic functions associated with complex polynomials

Two basic analytic functions α(z) and β(z) defined in domains depending on the location of the zeros of a complex polynomial P(z) are given by $\text{P′}\text{P}\text{=}\text{n}\text{(z ? α)}$ and P = (z ? β)ⁿ. These functions are studied with respect to their growth and their Laurent expansion coefficients. Applications to the location of zeros of complex polynomials are indicated.