PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 9, Pages 2667–2673 S 0002-9939(00)05520-9 Article electronically published on March 1, 2000 ON SOME PROPERTIES OF THE GAMMA FUNCTION ´ ARP ´ AD ELBERT AND ANDREA LAFORGIA (Communicated by Hal L. Smith) Abstract. Anderson and Qiu (1997) conjectured that the function log Γ(x+1) x log x is concave for x> 1. In this paper we prove this conjecture. We also study the monotonicity of some functions connected with the psi-function ψ(x) and derive inequalities for ψ(x) and ψ 0 (x). 1. Introduction For x> 0 let Γ(x) and ψ(x) denote the Euler’s gamma function, defined by Γ(x)= ∞ Z 0 e -t t x-1 dt, ψ(x)= Γ 0 (x) Γ(x) , respectively. There is a vast literature on these functions and a good reference to this can be found, for example, in the recent paper [2]. Anderson and Qiu showed that the function log Γ(x+1) x log x strictly increases from 1-γ to 1 as x increases from 1 to ∞, where γ =0.577... denotes the Euler-Mascheroni constant. To do this, they investigated the function f (x)= ψ 0 (1 + x)+ xψ 00 (1 + x) (1.1) and they found the representation f (x)= ∞ X n=1 n - x (n + x) 3 . (1.2) They proved, in a complicated way, that f (x) > 0 for x ∈ [1, 4) and formulated the following: Conjecture. The function log Γ(x+1) x log x is concave for x> 1. In Section 2 we extend the inequality f (x) > 0 from [1, 4) to (-1, ∞) (this extension is evident for -1 <x ≤ 1). Then we derive also new inequalities for ψ(x) and ψ 0 (x). In Section 3 we prove the conjecture formulated above by Anderson and Qiu [3]. Received by the editors October 23, 1998. 2000 Mathematics Subject Classification. Primary 33B15; Secondary 26A48, 26D07. Key words and phrases. Gamma function, psi function, monotonicity, inequalities. c 2000 American Mathematical Society 2667 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use