On Some Bounds for the Exponential Integral Function

Abstract

In 1934, Hopf established an elegant inequality bounding the exponential integral function. In 1959, Gautschi established an improvement of Hopf’s results. In 1969, Luke also established two inequalities with each improving Hopf’s results. In 1997, Alzer also established another improvement of Hopf’s results. In this paper, we provide two new proofs of Luke’s first inequality and as an application of this inequality, we provide a new proof and a generalization of Gautschi’s results. Furthermore, we establish some inequalities which are analogous to Luke’s second inequality and Alzer’s inequality. The techniques adopted in proving our results are simple and straightforward.