In this article, the monotonicities of two functions ¡ 1+ 1 x ¢ x+α and ¡1+α x ¢ x+β and their corresponding sequences ¡ 1+ 1 n ¢ n+α and ¡1+α n ¢ n+β are presented, and equivalent relations between the monotonicities of either these two functions or these two sequences are verified. As by-products, some new inequalities for the natural logarithm are obtained.