Convergence analysis of a Pad\'{e} family of iterations for the matrix sector function

The main purpose of this paper is to give a solution to a conjecture concerning a Pad\'{e} family of iterations for the matrix sector function that was recently raised by B. Laszkiewicz et al in [A Pad\'{e} family of iterations for the matrix sector function and the matrix $p$th root, Numer. Linear Algebra Appl. 2009; 16:951-970]. Using a sharpened version Schwarz's lemma, we also demonstrate a strengthening of the conjecture.