On Gautschi's harmonic mean inequality for the gamma function

[摘要] Let H-n = inf(xis an element ofSn) ((1)/(n) Sigma(k=1)(n) (1)/(Gamma(xk)))(-1), where S-n={(x(1),...x(n))is an element ofR(+)(n): Pi(k=1)(n) x(k) = 1}. Gautschi (SIAM J. Math. Anal. 5 (1974)) showed that H-2 = 1 and H-n < 1 for all n greater than or equal to 9. In this paper we prove his conjecture that H-n = 1 for n less than or equal to 8. (C) 2003 Elsevier B.V. All rights reserved.

[发布日期] 2003-08-01

[关键词] gamma function;harmonic mean;inequalities