O. Gomilko, D.B. Karp, M. Lin, K. Zietak,
Regions of convergence of a Pade family of iterations for the matrix sector function 
and the matrix p root


In this paper we prove a conjecture  on a common region of a convergence of Pade iterations 
for the matrix sector function.  For this purpose we show   that all  Pade approximants 
to a special case of  hypergeometric  function  have 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.  

keywors:  matrix sector function; matrix root; rational matrix iteration; 
Pade family of iterations; Pade approximant; hypergeometric function, hypergeometric identity