Estimates of majorizing sequences in the Newton-Kantorovich method: A further improvement

In the paper [F. Cianciaruso, E. De Pascale, Estimates of majorizing sequences in the Newton-Kantorovich method, submitted for publication] we proved that, ifξ:=aθbk⩽1(1+θθ1−θ)1−θ(θ1+θ)θ, then the following estimates for rn holdrn⩽(bk)−1θ(1+θθ1−θ)1−θθ(1−1(1+θ)n),∀n∈N. In the present paper we give a stronger (at least asymptotically) estimates on rn under a weaker condition on ξ. The techniques employed in the paper are similar to the ones used in [F. Cianciaruso, E. De Pascale, Estimates of majorizing sequences in the Newton-Kantorovich method, submitted for publication]. Finally, we make a comparison with previous results.