本文给出有关正项等差数列的两组优美的不等式,并给出它的一些应用.引理1(伯努利不等式)设1+α0,α≠0,n∈R且n≠0,1则有1)当0n1时(1+α)n1+nα;2)当n1或n0时(1+α)n1+nα.证略.引理2设正项等差数列{an}是单调递增的,其公差为d,α∈R,α≠0,1.则有:1)当α0时,1akαak+1 In this paper, we give two sets of graceful inequalities concerning positive exponential series and give some applications. Lemma 1 (Bernoulli inequality) Let 1+α0, α≠0, n∈R and n≠0, 1) When 0n1 (1+α)n1+nα; 2) When n1 or n0 (1+α)n1+nα. Proof. Lemma 2 Let the positive item series {an} be monotonous Incremented, the tolerance is d, α∈R, α≠0, 1. Then there are: 1) When α0, 1akαak+1