Inequalities And Asymptotic Expansions For The Gamma Function And Constant E

In this paper,we prove some asymptotic expansions and inequalities for special function,the main results are as follows:1.Based on the Pade approximation method,we determine the coefficients ai and bj(1≤j≤k)such that where k≥1 is any given integer.Based on the obtained result,we establish new upper bounds for(1+1/x)x.As an application,we give a generalized Carieman-type inequality.2.We give an asymptotic expansion for the gamma function.Based on the obtained result,we establish inequality for the gamma function,for x≥2,we have3.We give some asymptotic expansions for the ratio of gamma functions.Based on the obtained result,we establish some inequalities for the ratio of gamma functions,for x>3,we have where and For every n ∈N,we haveπ/2 exp(μ(n))<Wn<π/2exp(v(n)),where and (?)...