Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely monotonic functions related to gamma function

An accurate approximation formula for gamma function

“…More results involving the approximation formulas for the factorial or gamma function can be found in [9,[11][12][13][14][15][16][17][18][19][20][21][22][23] and the references cited therein. Several nice inequalities between gamma function and the truncations of its asymptotic series can be found in [24,25]. Now let us focus on the Windschitl's approximation formula (see [6,Eq.…”

Notes on a Double Inequality for Ratios of any Two Neighbouring Non-zero Bernoulli Numbers

“…is strictly increasing (or strictly decreasing, respectively) if and only if 1 θ ≤ − (or 0 θ ≥ , respectively). The double inequality (2) has been cited and applied in the papers [22][23][24][25][26][27][28][29].…”

Windschitl type approximation formulas for the gamma function

“…For this reason, scholars strive to find various better approximations for the factorial or gamma function by using different ideas and techniques, for instance, Ramanujan [ 1 , p. 339], Burnside [ 2 ], Gosper [ 3 ], Alzer [ 4 ], Shi et al [ 5 ], Batir [ 6 , 7 ], Mortici [ 8 – 12 ], Nemes [ 13 , Corollary 4.1], [ 14 ], Qi et al [ 15 , 16 ], Feng and Wang [ 17 ], Chen [ 18 – 21 ], Yang et al [ 22 – 25 ], Lu et al [ 26 – 28 ], Xu et al [ 29 ]. Some properties of the remainders of certain approximations for the gamma function can be found in [ 4 , 16 , 23 , 30 – 35 ].…”