On Gautschiʹs harmonic mean inequality for the gamma function

LetHn=infx∈Sn1n ∑k=1n 1Γ(xk)−1,where Sn={(x1,…,xn)∈R+n: ∏k=1n xk=1}. Gautschi (SIAM J. Math. Anal. 5 (1974)) showed that H2=1 and Hn<1 for all n⩾9. In this paper we prove his conjecture that Hn=1 for n⩽8.