Approximation for constant e and its applications

Estimating the digamma and trigamma functions by completely monotonicity arguments

“…In this paper, we study the complete monotonicity of the functions F a , then we apply these results to obtain new sharp bounds for the digamma and trigamma functions. The problem of estimating the gamma and polygamma functions has attracted the attention of many researchers, since they are close related to the theory of zeta functions [1,8,12,16,27,39], multiple gamma and related functions [7,[9][10][11]13,[24][25][26]30,33,35,38,40], gamma type distributions [15,29], or harmonic sums [23,34]. There are also many recent investigations dealing with one-sided and two-sided inequalities involving the digamma, trigamma, polygamma and other related functions, see e.g., [5,6,14,31,32].…”

New improvements of the Stirling formula

“…In consequence, it has been deeply studied by many authors due to its basic role in the theory of the gamma function and related functions. See, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][17][18][19][20][21][22][23][24][25][26][27][28][29][31][32][33][34][35][36][37] and all references therein. We use here the discrete representation The superiority of these formulas over the Stirling formula follows by a subsequent inequality (2.2).…”

Some Refinements and Generalizations of I. Schur Type Inequalities

“…It is well known that x n = (1 + (1/ n )) n and y n = (1 + (1/ n )) n +1 are, respectively, monotone increasing and monotone decreasing, and both of them converge to the constant e . In fact, extensive researches for the estimated value of e have been studied [ 1 – 4 ], and the methods for estimating the value of e are of benefit to the improvements of the Hardy inequality, Carleman inequality, Gamma function inequality, and so forth [ 5 – 13 ], which is an essential motivation for this work. Klambauer and Schur have reached the following conclusion.…”