Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article One Method for Proving Inequalities by Computer | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 78691 8 pages doi 2007 78691 Research Article One Method for Proving Inequalities by Computer Branko J. Malesevic Received 31 August 2006 Revised 30 October 2006 Accepted 31 October 2006 Recommended by Andrea Laforgia We consider a numerical method for proving a class of analytical inequalities via minimax rational approximations. All numerical calculations in this paper are given by Maple computer program. Copyright 2007 Branko J. Malesevic. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Some particular inequalities In this section we prove two new inequalities given in Theorems and . While proving these theorems we use a method for inequalities of the form f x 0 for the continuous function f a fi - R. . Let us consider some inequalities for the gamma function which is defined by the integral T z f e-ttz-1dt 0 which converges for Re z 0. In 1 the following statement is proved. Lemma . For x e 0 1 the following inequalities are true r x 1 x2 - x 1 x 2 r x 1 5. 2 Journal of Inequalities and Applications The previous statement 1 Lemma is proved by the approximative formula for the gamma function r x 1 by the polynomial of the fifth order P5 x - - 1 which has the numerical bound of the absolute error e 5 10-5 for values of argument x e 0 1 2 Formula page 257 . In the Maple computer program we use numapprox package 3 for obtaining the minimax rational approximation R x Pm x Qn x of the continuous function f x over segment a fi m is the degree of the polynomial Pm x and n is the degree of the polynomial Qn x . Let e x f x - R x be the error function of an approximation over segment a fi . Numerical computation
