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Let f : B ( x 0 , R ) ⊆ X → Y be an operator, with X and Y Banach spaces, and f ′ be Hölder continuous with exponent θ. The convergence of the sequence of Newton–Kantorovich approximations is a classical tool to solve the equation f ( x ) = 0. The convergence of x n is often reduced to the study of the majorizing sequence r n defined by with a, b, k parameters related to f and f ′. We extend an estimate for r n , known in the Lipschitz case, to the Hölder case. The proof requires the introduction of a multiplicative factor in the sequence estimating r n , estimates of the ratio , and the use of two parallel induction processes on the sequences r n and . In the last section, we make a comparison with our previous results. [ABSTRACT FROM AUTHOR]