Some improvements of one method for proving inequalities by computer

In the article [17] we consider only the case when n and m are non-negative integer points determined by: n ≥ 1 is the multiplicity of the root x = a, otherwise n = 0 if x = a is not the root; and m ≥ 1 is the multiplicity of the root x = b, otherwise m = 0 if x = b is not the root. In this case, if for the function f(x) at the point x = a there is an approximation of the function by Taylor polynomial of n-th order and at point x = b there is an approximation of the function by Taylor polynomial of m-th order, then [17]: