Asymptotic Expansions And Inequalities For Special Functions

In this paper,we prove the inequalities and asymptotic expansions for the Psi function,Euler–Mascheroni constant and Landau constants.The main results are as follows:1.Continued fraction estimates for the Psi function with application:?a?For x > 0,Based on this expansion,we obtain the following.For all integers n?1,letthenwe obtain the higher order estimate for the Euler–Mascheroni constant.2.Inequality and asymptotic expansion for the Euler–Mascheroni constant:?a?For all integers n ? 1,let the sequence Pnbe defined by?2.1.3?.Thenwith the best possible constants?b?The harmonic number has the following asymptotic expansion:with the coefficients ajgiven by the recursive formulawhere R_jare given by?c?Let P_nbe defined by?2.1.3?.ThenIn the second chapter,we give a recurrence relation for determining the coefficients of asymptotic expansion for the harmonic number.We establish inequalities for the Euler–Mascheroni constant.3.Continued fraction estimates for the Landau constants:?a?For n ? N₀,where?b?For n ? N₀,where?c?For n ? N₀,In the third chapter,we present continued fraction estimates for the Landau constants.We obtain the better approximation formulas for the Landau constants.