Inequalities for Zero-Balanced Hypergeometric Functions

Abstract

Abstract.The authors study certain monotoneity and convexity properties of the Gaussian hypergeometric function and those of the Euler gamma function.

On some inequalities for the gamma and psi functions

“…Anderson et al [3] used the monotonicity of f 1 to prove that the function g 1 (x) = x 1/2 (e/x) x Γ(x) is decreasing on (0, ∞), and that g 2 (x) = x(e/x) x Γ(x) is increasing on (0, ∞). The following theorem provides a slight extension of these results.…”

“…There exists a very extensive literature on the gamma function. In particular, numerous remarkable inequalities involving Γ and its logarithmic derivative ψ = Γ /Γ have been published by different authors; see, e.g., [2], [3], [6], [7], [9], [12], [13], [18]- [27], [29]- [33], [35]- [46], [50]. Many of these inequalities follow immediately from the monotonicity properties of functions which are closely related to Γ and ψ.…”

“…In a recently published article G. D. Anderson et al [3] proved that the function f (x) = x(log(x) − ψ(x)) is strictly decreasing and strictly convex on (0, ∞). Moreover, the authors presented (complicated) proofs for…”

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

“…is strictly decreasing for u > 0 (see Theorem 3.2 (2) of Anderson et al (1995)), and by the definition of (2),…”

“…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…”