Completely monotonic functions and inequalities associated to some ratio of gamma function

Completely monotonic functions and inequalities associated to some ratios of the Pochhammer k-symbol

“…2 Remark 1. When k = 3, i = 2 or k = 3, i = 1 in Theorems 1 and 2, we can obtain Theorem 1 or 2 and Corollary 1 or 2 of Mortici et al [11], respectively. So, our results improve and generalize the corresponding conclusions of the above paper.…”

“…and R n 2,3 = 1·4···(3n−2) 3·6···(3n) , which were studied by Mortici et al [11], respectively. The main contributions of this article are to prove that these functions in (7) are completely monotonic on (0, ∞) and to establish some sharp inequalities.…”

“…2 Remark 2. Clearly, the left sides of Theorems 3 and 4 of Mortici et al [11] are included in (22), but the right sides can not be included in (23). Despite all this, (22) and (23) give a unified and fast estimation method.…”

Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function

“…For more history of π see [6,7,14]. Very recently, Mortici et al [36] proved some completely monotonic functions and inequalities associated with the ratio of gamma functions. x/2 (4.4) gives at least eight decimal places of the gamma function.…”

New Inequalities for Gamma and Digamma Functions

“…In the past different papers appeared providing inequalities for the gamma, digamma, and polygamma functions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). By using the mean value theorem to the function log Γ( ) on [ , + 1], with > 0 and > 0, Batir [19] presented the following inequalities for the gamma and digamma functions: …”