MATHEMATICS OF COMPUTATION Volume 74, Number 250, Pages 723–742 S 0025-5718(04)01675-8 Article electronically published on May 18, 2004 SOME PROPERTIES OF THE GAMMA AND PSI FUNCTIONS, WITH APPLICATIONS S.-L. QIU AND M. VUORINEN Abstract. In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume Ωn of the unit ball B n ⊂ R n , the surface area ω n-1 of the unit sphere S n-1 , and some related constants. 1. Introduction For real and positive values x and y, the Euler gamma function, the beta and psi (or polygamma) functions are defined as Γ(x)=  ∞ 0 t x-1 e -t dt, B(x, y)= Γ(x)Γ(y) Γ(x + y) , ψ(x)= Γ ′ (x) Γ(x) , (1.1) respectively. For the extensions to complex variables and for the basic properties of these functions, see [AS], [AAR], [Mi], [T], and [WW]. Over the past half century, many authors have obtained various properties and inequalities for these very important functions (see [A1], [A2], [A5], [G], [Ke], [K2], [L], [MSC] and bibliographies therein). Formulas for the volumes of geometric bodies sometimes involve the gamma function. This topic and related inequalities have been studied recently in [AQ], [A1]–[A4], [BP], [EL]. Let B n and S n-1 be the unit ball and unit sphere in R n , respectively, Ω n be the volume of B n , and ω n-1 denote the surface area of S n-1 . Set Ω 0 = 1. It is well known that Ω n is increasing for 2 ≤ n ≤ 5 and decreasing for n ≥ 5, while ω n-1 is increasing for 2 ≤ n ≤ 7 and decreasing for n ≥ 7 (see [BH, pp. 263–264] and [AVV1, p. 38]): Ω n = 2π n Ω n-2 = π n/2 Γ(1 + n/2) , ω n-1 = nΩ n =2 π n/2 Γ(n/2) . (1.2) Throughout this paper, we let γ =0.57721 ··· = -ψ(1) denote the Euler- Mascheroni constant, (2n-1)!! = (2n-1)(2n-3) ··· 1 for n ∈ N = {k; k is a natural number}, I n =  π/2 0 sin n-2 tdt, J n =  π/2 0 sin (2-n)/(n-1) tdt, (1.3) Received by the editor April 2, 2002 and, in revised form, September 27, 2003. 2000 Mathematics Subject Classification. Primary 33B15; Secondary 26B15, 26D15, 51M25. Key words and phrases. Gamma function, beta function, psi function, monotoneity, concavity, inequalities. c 2004 American Mathematical Society 723 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use