## A completely monotonic function involving divided differences of psi and polygamma functions and an application

Venue: | RGMIA Res. Rep. Coll |

Citations: | 17 - 13 self |

### BibTeX

@ARTICLE{Qi_acompletely,

author = {Feng Qi and Bai-ni Guo},

title = {A completely monotonic function involving divided differences of psi and polygamma functions and an application},

journal = {RGMIA Res. Rep. Coll},

year = {}

}

### OpenURL

### Abstract

Abstract. A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw’s double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw’s double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in R n−1 and R n respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized. 1.

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Citation Context ...] e −xu du, (26) ∫ ∞ As x → ∞, ψ (n) (x) = (−1) n+1 0 u n 1 − e −ue−xu du, (27) ψ (n−1) (x + 1) = ψ (n−1) (x) + (−1)n−1 (n − 1)! x n , (28) lnx − 1 1 < ψ(x) < lnx − . (29) x 2x ψ ′ (x) ∼ 1 x Lemma 3 (=-=[37, 38]-=-). For s > r > 0, 1 + + · · · . (30) 2x2 exp[(s − r)ψ(s)] > Γ(s) > exp [(s − r)ψ(r)] . (31) Γ(r) Lemma 4 ([57]). A product of finite completely monotonic functions is also completely monotonic. 3. Pro... |

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Citation Context ...s) 1 1 − . (6) 4 2 This suggests us to introduce a function ⎧[ ⎪⎨ Γ(x + t) zs,t(x) = Γ(x + s) ⎪⎩ eψ(x+s) − x, ] 1/(t−s) − x, s ̸= t s = t on x ∈ (−α, ∞) for real numbers s and t and α = min{s, t}. In =-=[13, 22, 34, 35, 36, 39, 51, 52]-=-, the monotonic and convex properties of zs,t(x) were established by using Laplace transform and other complicated techniques. Their basic calculation is as follows: z ′′ s,t (x) = [zs,t(x) + x] = zs,... |

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Citation Context ...⎩ · (n − 1)!! n!! (n − 1)!! n!! for n even, for n odd, where n!! denotes the double factorial. It has been estimated by many mathematicians and a lot of inequalities were established in, for example, =-=[12, 14, 15, 16, 17, 18, 19, 20, 25, 27, 43, 54, 58]-=- and related references therein. The third aim of this paper is, by utilizing Theorem 2, to prove two sharp double inequalities relating to Wallis cosine or sine formula (15) and to bound the probabil... |

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