  In the framework of the majorization technique, an improved condition is
proposed for the semilocal convergence of the Newton method under the mild
assumption that the derivative of the involved operator F(x) is continuous. Our
starting point is the Argyros representation of the optimal upper bound for the
distance between the adjacent members of the Newton sequence. The major novel
element of our proposal is the optimally reconstructed 'first integral'
approximation to the recurrence relation defining the scalar majorizing
sequence. Compared to the previous results of Argyros, it enables one to obtain
a weaker convergence condition that leads to a better bound on the location of
the solution of the equation F(x)=0 and allows for a wider choice of initial
guesses. In the simplest case of the Lipschitz continuous derivative operator,
an explicit restriction is found which guarantees that the new convergence
condition improves the famous Kantorovich condition.
