The well-known Kantorovich theorem, concerning the existence and uniqueness of a solution \(x^*\) of a nonlinear equation in a Banach space as well as the convergence of the Newton process to this solution, gives also an estimate for the error of the n-th iterate \(x_ n\), i.e. for the quantity \(\| x*-x_ n\|\). Later there are given, by many authors, improved versions of this estimate, derived with the use of different techniques. The present author considers four of these improvements, one given by \textit{J. E. Dennis} jun. [SIAM J. Numer. Anal. 6, 493-507 (1969; Zbl 0221.65098))] and \textit{R. A. Tapia} [Am. Math. Mon. 78, 389-392 (1971; Zbl 0215.274)], two by \textit{W. B. Gragg} and \textit{R. A. Tapia} [SIAM J. Numer. Anal. 11, 10-13 (1974; Zbl 0284.65042)], one by \textit{F. A. Potra} and \textit{V. Pták} [Numer. Math. 34, 63-72 (1980; Zbl 0434.65034)]. He gives a unified derivation for all these estimates using only the Kantorovich theorem and his recurrence relations. Finally the author puts these four error bounds in order according to their accuracy.