Monotonicity Properties of the Gamma Function

Abstract

Let Gc(x) = log Gamma(x) - x log x + x- 1/2log(2 pi) + 1/2 psi(x + c) (x > 0; c >= 0). We prove that G(a) is completely momotonic on (0, infinity) if and only a >= 1/3. Also, -G(b) is completely monotonic on (0, infinity) if and only if b = 0. An application of this result reveals that the best possible nonnegative constants alpha, beta in root 2 pi x(x) exp (-x - 1/2 psi(x + alpha) < Gamma (x) < root 2 pi x(x) exp (-x - 1/2 psi(x + beta) (x > 0) are given by alpha = 1/3 and beta + 0. (c) 2006 Elsevier Ltd. All rights reserved.