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We present a new method for the computation of the solutions of nonlinear equations when it is necessary to use high precision. We improve the Euler-Chebyshev iterative method which is a generalization of an improvement of Newton's method. A symbolic computation allows us to find the best coefficients respect to the local order of convergence. The adaptation of the strategy presented here gives an additional iteration function with an additional evaluation of the function. It provides a lower cost if we use adaptive multiprecision arithmetics. The numerical results computed using this system, with a floating point representing a maximum of 210 decimal digits, support this theory.  2005 Elsevier Inc. All rights reserved.

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a b s t r a c tIn this paper, splitting finite sums, refinements of the inequalities of Aczél, Popoviciu and Bellman are obtained.