## Some monotonicity properties of gamma and q-gamma functions, Available onlie at http://arxiv.org/abs/0709.1126v2

Citations: | 3 - 0 self |

### BibTeX

@MISC{Gao_somemonotonicity,

author = {Peng Gao},

title = {Some monotonicity properties of gamma and q-gamma functions, Available onlie at http://arxiv.org/abs/0709.1126v2},

year = {}

}

### OpenURL

### Abstract

Abstract. We prove some properties of completely monotonic functions and apply them to obtain results on gamma and q-gamma functions. 1.

### Citations

2474 | Stegun (Eds.), Handbook of mathematical functions: With formulas, graphs, and mathematical tables - Abramowitz, A - 1964 |

44 | On some inequalities for the gamma and psi functions
- Alzer
- 1997
(Show Context)
Citation Context ...n, for positive x and 0 < q < 1, 0 t x e −tdt t . Γ 1/q(x) = q (x−1)(1−x/2) Γq(x), we see that limq→1 Γq(x) = Γ(x). For historical remarks on gamma and q-gamma functions, we refer the reader to [26], =-=[2]-=- and [5]. There exists an extensive and rich literature on inequalities for the gamma and q-gamma functions. For the recent developments in this area, we refer the reader to the articles [22], [2], [5... |

31 |
Some extensions of W. Gautschi’s inequalities for the gamma function
- KERSHAW
- 1983
(Show Context)
Citation Context ...lso note that ψ(x) is an increasing function on (0,+∞) and √ (x + 1)(x + s) ≥ x + s1/2 , we see that the inequalities in (3.2) refine the case q → 1 in Theorem 3.3 and the following result of Kershaw =-=[25]-=-, which states that for positive x and 0 ≤ s ≤ 1, exp ( (1 − s)ψ(x + s 1/2 ) ) ≤ Γ(x + 1) ≤ exp Γ(x + s) ( 1 + s (1 − s)ψ(x + 2 ) We now show the lower bound above and the corresponding one in (3.1) a... |

27 | Some gamma function inequalities - Alzer - 1993 |

21 |
On gamma function inequalities
- Bustoz, Ismail
- 1986
(Show Context)
Citation Context ...eorem now follows by applying Lemma 2.3 to the functions f1(x)/f1(x + 1/2) and f2(x)/f2(x + 1/2). □ We remark here the above corollary is essentially Theorem 4 and 5 in [20], except that Theorem 4 of =-=[13]-=- is originally stated as the function x ↦→ (1 − 1 2x )−1/2 Γ2 (x + 1/2) Γ(x)Γ(x + 1) is completely monotonic on (1/2,+∞). One can also obtain this result by modifying our approach above. As another ap... |

20 |
Sharp inequalities for the digamma and polygamma functions
- Alzer
(Show Context)
Citation Context ...nd [5]. There exists an extensive and rich literature on inequalities for the gamma and q-gamma functions. For the recent developments in this area, we refer the reader to the articles [22], [2], [5]-=-=[7]-=-, [29] and the references therein. Many of these inequalities follow from the monotonicity properties of functions which are closely related to Γ (resp. Γq) and its logarithmic derivative ψ (resp. ψq)... |

17 | The best bounds in Gautschi’s inequality - Elezović, Giordano, et al. |

16 | Completely monotonic functions associated with the Gamma function and its q-analogues - ISMAIL, LORCH, et al. - 1986 |

16 |
functions as limit cases of q-ultraspherical polynomials
- Koornwinder, Jacobi
- 1990
(Show Context)
Citation Context ...tion is defined for positive real numbers x and q ̸= 1 by Γq(x) = (1 − q) 1−x ∞∏ 1 − qn+1 1 − qn+x, 0 < q < 1; ∞∏ n=0 Γq(x) = (q − 1) 1−x q 1 2 x(x−1) n=0 1 − q−(n+1) 1 − q−(n+x), q > 1. We note here =-=[26]-=- the limit of Γq(x) as q → 1 − gives back the well-known Euler’s gamma function: lim q→1− Γq(x) = Γ(x) = ∫ ∞ Note also from the definition, for positive x and 0 < q < 1, 0 t x e −tdt t . Γ 1/q(x) = q ... |

15 |
Completely monotonic functions involving the gamma and q-gamma functions
- Grinshpan, Ismail
(Show Context)
Citation Context ...onotonic on (a,b) if it has derivatives of all orders and (−1) k f (k) (x) ≥ 0,x ∈ (a,b),k ≥ 0. Lemma 2.4 below asserts that f(x) = e −h(x) is completely monotonic on an interval if h ′ is. Following =-=[20]-=-, we call such functions f(x) logarithmically completely monotonic. We note here that limq→1 ψq(x) = ψ(x) (see [27]) and that ψ ′ and ψ ′ q are completely monotonic functions on (0,+∞) (see [23], [7])... |

14 |
Mean-value inequalities for the polygamma functions
- Alzer
(Show Context)
Citation Context ...on −Fp,m,n,q(x;dp,m,n,q) is also completely monotonic on (0,+∞) when q > 0, .10 PENG GAO Proof. We first prove the assertion for Fp,m,n,q(x;cp,m,n,q) with q ≥ 1 and the proof here uses the method in =-=[10]-=-. Using the integral representation (2.3) for (−1) n+1 ψ (n) (x) and using ∗ for the Laplace convolution, we get where g(t) = = ∫ ∞ Fp,m,n,q(x;cp,m,n,q) = 0 0 e −xt g(t)dt, tm t ∗ 1 − e−t n t − cp,m,n... |

13 |
Inequalities and monotonicity properties for gamma and q-gamma functions, in
- ISMAIL, MULDOON
- 1994
(Show Context)
Citation Context ... to [26], [2] and [5]. There exists an extensive and rich literature on inequalities for the gamma and q-gamma functions. For the recent developments in this area, we refer the reader to the articles =-=[22]-=-, [2], [5]-[7], [29] and the references therein. Many of these inequalities follow from the monotonicity properties of functions which are closely related to Γ (resp. Γq) and its logarithmic derivativ... |

13 |
Some inequalities for the Gamma function
- KEČKIĆ, VASIĆ
- 1971
(Show Context)
Citation Context ...1/2) = 1, f(x + a;1) f(x;1) We can recast the above as bb−1 aa−1ea−b ≤ Γ(b) bb−1/2 ≤ Γ(a) aa−1/2 ea−b , b > a > 0. f(x + a;1) ≥ lim = 1. x→+∞ f(x;1) The above inequalities are due to Kečkić and Vasić =-=[24]-=- for the case b ≥ a > 1. We now look at some functions involving the q-gamma function. Ismail et al. [21, Theorem 2.2] proved that for q ∈ (0,1) the function (1 − q) x Γq(x) is logarithmically complet... |

11 | Some new inequalities for gamma and polygamma functions - Batir - 2005 |

9 |
Convexity, Schur-convexity and bounds for the Gamma function involving the Digamma function
- Merkle
(Show Context)
Citation Context ...x + 1) + ψq(x + s) 2 ) ≤ Γq(x ( + 1) ≤ exp (1 − s)ψq Γq(x + s) ( 1 + s) x + 2 ) . The upper bound in Theorem 3.3 is due to Ismail and Muldoon [22]. Our proof here is similar to that of Corollary 3 in =-=[28]-=-, which asserts for positive x, (1 − s) ( ) Γ(x + 1) 1 + s (3.1) ψ(x + 1) + ψ(x + s) < ln < (1 − s)ψ(x + ), 0 < s < 1. 2 Γ(x + s) 2 We further note the following integral analogue of Theorem 3.5 in [7... |

8 |
Inequalities for the volume of unit ball in R n
- ALZER
(Show Context)
Citation Context ...me of our results in the previous sections to study inequalities for the volume Ωn of the unit ball in Rn : π Ωn = n/2 Γ(1 + n/2) . There exists many inequalities involving Ωn, we refer the reader to =-=[3]-=- and the references therein. In [3], Alzer has shown that for n ≥ 2, 2n + 1 4π < ( ) 2 Ωn−1 Ωn Ωn ≤ 2n + π − 2 . 4π The above results were improved by him recently [7, Theorem 3.8] to be: 1 (6.1) 2π e... |

8 |
Sharp bounds for the ratio of q−Gamma functions
- ALZER
(Show Context)
Citation Context ...ositive x and 0 < q < 1, 0 t x e −tdt t . Γ 1/q(x) = q (x−1)(1−x/2) Γq(x), we see that limq→1 Γq(x) = Γ(x). For historical remarks on gamma and q-gamma functions, we refer the reader to [26], [2] and =-=[5]-=-. There exists an extensive and rich literature on inequalities for the gamma and q-gamma functions. For the recent developments in this area, we refer the reader to the articles [22], [2], [5]-[7], [... |

7 |
A subadditive property of the gamma function
- ALZER, RUSCHEWEYH
(Show Context)
Citation Context ...y monotonic on (0,+∞). Similarly, one shows that (x) − f ′′(x + 1) is completely monotonic on (0,+∞) and this completes the proof. □ that f ′′ 1/2 f ′′ 0 0 We note here that recently, Alzer and Batir =-=[10]-=- showed that the following function (x > 0,c ≥ 0) Gc(x) = ln Γ(x) − xln x + x − 1 1 ln(2π) + ψ(x + c) 2 2 is completely monotonic if and only if c ≥ 1/3 and −Gc(x) is completely monotonic if and only ... |

6 | Grinshpan, Inequalities for the gamma and q−gamma functions - Alzer, Z |

6 | Inequalities involving gamma and psi function
- CLARK, ISMAIL
(Show Context)
Citation Context ...graph below Lemma 2.2) and the ring structure of completely monotonic functions that (−1) n+1 x n ψ (n) (x) is completely monotonic on (0,+∞). This will further imply a conjecture of Clark and Ismail =-=[15]-=-, which asserts that the n-th derivative of (−1) n+1 x n ψ (n) (x) is completely monotonic on (0,+∞) and this has been recently disproved by Alzer et al. [8].that SOME MONOTONICITY PROPERTIES OF GAMM... |

6 | Summations for basic hypergeometric series involving a q-analogue of the Digamma function
- Krattenthaler, Srivastava
- 1996
(Show Context)
Citation Context ...erts that f(x) = e −h(x) is completely monotonic on an interval if h ′ is. Following [20], we call such functions f(x) logarithmically completely monotonic. We note here that limq→1 ψq(x) = ψ(x) (see =-=[27]-=-) and that ψ ′ and ψ ′ q are completely monotonic functions on (0,+∞) (see [23], [7]). Thus, one expects to deduce results on gamma and q-gamma functions from properties of (logarithmically) completel... |

6 | M.,2005. Some properties of the gamma and psi functions, with applications
- Qiu, Vuorinen
(Show Context)
Citation Context ...]. There exists an extensive and rich literature on inequalities for the gamma and q-gamma functions. For the recent developments in this area, we refer the reader to the articles [22], [2], [5]-[7], =-=[29]-=- and the references therein. Many of these inequalities follow from the monotonicity properties of functions which are closely related to Γ (resp. Γq) and its logarithmic derivative ψ (resp. ψq). Here... |

5 | Monotonicity and convexity for the gamma function - Chen |

3 |
On a conjecture of
- Alzer, Berg, et al.
(Show Context)
Citation Context ...er imply a conjecture of Clark and Ismail [18], which asserts that the n-th derivative of (−1) n+1 x n ψ (n) (x) is completely monotonic on (0,+∞) and this has been recently disproved by Alzer et al. =-=[11]-=-. Using the integral representation for (−1) n+1 ψ (n) (x) in (2.3), we obtain via integration by parts that (−1) n+1 xψ (n) ∫ ∞ (x) = 0 tn 1 − e−td(−e−xt ) = −e−xttn 1 − e−t ∣ +∞ 0 + ∫ ∞ Note that dm... |

3 | Differential and integral f-means and applications to digamma function - Elezović, Pečarić |

2 | A note on Schur-convex functions - Elezović, Pečarić |

2 |
Some characterizations of q-factorial functions
- Kairies, Muldoon
- 1982
(Show Context)
Citation Context ...owing [20], we call such functions f(x) logarithmically completely monotonic. We note here that limq→1 ψq(x) = ψ(x) (see [27]) and that ψ ′ and ψ ′ q are completely monotonic functions on (0,+∞) (see =-=[23]-=-, [7]). Thus, one expects to deduce results on gamma and q-gamma functions from properties of (logarithmically) completely monotonic functions, by applying them to functions related to ψ ′ or ψ ′ q . ... |

1 | A power mean inequality for the gamma function - Alzer |

1 | A note on the volume of sections of B n p - Gao |

1 | and superadditive properties of Euler’s gamma function - Alzer, Sub- |