Improvement of Newman inequality ✩

Let a = exp (−1/√n). Newman inequality is n−1 k=1 1− ak 1+ ak < e−√n, ∀n 5. We prove in this paper that s−1 k=1 1−ak 1+ak = n 1 4 e−π2 4 √n+O(1), ∀s n, which will be applied to improve the estimate concerning the approximation of |x| by using Newman’s construction.  2005 Elsevier Inc. All rights reserved.