Zero-free disks in families of analytic functions

Abstract

This note is concerned with families of functions f(z,q), analytic in both the variable z∈ℂ and the parameter q∈ℂ. In the one-parameter subfamily characterized by q=e^iϑ, ϑ∈ℝ we consider the problem of finding the zero z=z_m of f(z,e^iϑm) with minimum modulus. An algorithm for calculating ϑ_m, z_m based on Newton’s method will be described. The problem arises in several fields of approximation theory, notably in connection with certain Padé approximants and with the Whittaker and the power series constants. Conjectured values of these constants will be given in extended precision.