A class of logarithmically completely monotonic functions related to (1 + 1/x) x and an application 1

In this paper, sufficient and necessary conditions such that the function [ e/(1 + 1/x)x+β ]x+α is completely monotonic or logarithmically completely monotonic in (0,∞) are established. As byproducts, several inequalities related to (1 + 1/x)x are obtained and one of them is used to strengthen van der Corput’s inequality. 2000 Mathematics Subject Classification: Primary 26A48, 26A51; Secondary 26D15

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