Regions of convergence of a Padé family of iterations for the matrix sector function and the matrix pth root

In this paper, we prove a conjecture on a common region of a convergence of Padé iterations for the matrix sector function. For this purpose, we show that all Padé approximants to a special case of hypergeometric function have a power series expansion with positive coefficients. Using a sharpened version of Schwarz's lemma, we also demonstrate a better estimate of the convergence speed. Our results are also applicable to a family of rational iterations for computing the matrix pth root.