On a generalization of an upper bound for the exponential function (Q1014652)

In this paper, a generalization of an upper bound for the exponential function is obtained. Let \[ U(n,x)=1-\frac{1}{n}+\frac{1}{n}(\frac{1+(1-\frac{1}{n})x}{1-\frac{x}{n}})^{n}, \] and \[ \frac{1}{n-1}(U(n,x)-(1-x)^{-1})=\sum_{k=2}^{\infty}\frac{P_{k}(n)x^{k}}{k!n^{k-1}}. \] The new results are obtained: (1) \(\exp(\frac{n(x-1)}{n+x-1})\leq\frac{n-1+x^{n}}{n}\), for \(n=p/q, p>2q>0, x>-1\), or \(p<2q, x>(1-p/q)^{1/q}\) (\(p,q\) are odd integers); (2) \(P_{k}(n)\) is a monic polynomial of degree \(2k-3\).