On some inequalities for the  gamma and psi functions

Alzer, Horst

doi:10.1090/s0025-5718-97-00807-7

Cited by 359 publications

(193 citation statements)

References 47 publications

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“…are strictly decreasing for z > 0 (see Theorem 10 of Alzer (1997)). The bias follows from Lemma A.1 (ii); note…”

Section: A2 Proofs Of Theorems 1-4 and Remarkmentioning

confidence: 99%

“…using Lemma A.2 (i) and noting that, given p > 0, Γ(z)Γ(z + 2p)/Γ 2 (z + p) is strictly decreasing for z > 0 (see Theorem 10 of Alzer (1997)). Also, we obtain…”

Section: Proof Of Remark 2 (I) Remark 1 (I) and Lemma A2 (I) Yieldmentioning

confidence: 99%

“…Γ(z)Γ(z + p 1 + p 2 )/{Γ(z + p 1 )Γ(z + p 2 )}, is strictly decreasing for z > 0 (see Theorem 10 of Alzer (1997)). Alzer (1997;page 386)). …”

Section: Appendix B: Proof Of Lemma A4mentioning

confidence: 99%

“…Now, we know that, given p > 0, Γ(z)Γ(z + 2p)/Γ 2 (z + p) is strictly decreasing for z > 0 (see Theorem 10 of Alzer (1997)), and that z 1−z e z Γ(z) > 1 is strictly increasing for z > 0 (see Theorem 3.2 (2) of Anderson et al (1995)). Then, for any ρ ≥ c and γ ̸ = 0 (in this case, α γ (ρ) + q − 1/γ > 0), we use (A1) and Lemma B.2 to get…”

mentioning

confidence: 99%

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Limiting bias-reduced Amoroso kernel density estimators for non-negative data

Igarashi

Kakizawa

2017

Communications in Statistics - Theory and Methods

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The Amoroso kernel density estimator ) for nonnegative data is boundary-bias-free and has the mean integrated squared error (MISE) of order O(n −4/5 ), where n is the sample size. In this paper, we construct a linear combination of the Amoroso kernel density estimator and its derivative with respect to the smoothing parameter. Also, we propose a related multiplicative estimator. We show that the MISEs of these bias-reduced estimators achieve the convergence rates n −8/9 , if the underlying density is four times continuously differentiable. We illustrate the finite sample performance of the proposed estimators, through the simulations.

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“…are strictly decreasing for z > 0 (see Theorem 10 of Alzer (1997)). The bias follows from Lemma A.1 (ii); note…”

Section: A2 Proofs Of Theorems 1-4 and Remarkmentioning

confidence: 99%

“…using Lemma A.2 (i) and noting that, given p > 0, Γ(z)Γ(z + 2p)/Γ 2 (z + p) is strictly decreasing for z > 0 (see Theorem 10 of Alzer (1997)). Also, we obtain…”

Section: Proof Of Remark 2 (I) Remark 1 (I) and Lemma A2 (I) Yieldmentioning

confidence: 99%

“…Γ(z)Γ(z + p 1 + p 2 )/{Γ(z + p 1 )Γ(z + p 2 )}, is strictly decreasing for z > 0 (see Theorem 10 of Alzer (1997)). Alzer (1997;page 386)). …”

Section: Appendix B: Proof Of Lemma A4mentioning

confidence: 99%

mentioning

confidence: 99%

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Limiting bias-reduced Amoroso kernel density estimators for non-negative data

Igarashi

Kakizawa

2017

Communications in Statistics - Theory and Methods

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show abstract

“…For example, they play a role in complex analysis, number theory, potential theory, probability theory [9], physics [20], numerical and asymptotic analysis, integral transforms [41], and combinatorics. Some related references are listed in [2,3,4,5,6,15,23,41]. In recent years, inequalities and (logarithmically) completely monotonic functions involving the gamma, psi, or polygamma functions are established by some mathematicians (see [2,3,4,11,15,17,21,31] and related references therein).…”

Section: ∞) This Tells Us That F ∈ C[[0 ∞)] If and Only If It Is mentioning

confidence: 99%