Finbarr Holland showed in the case that \( p_1=\cdots=p_k=1/k \) that expanding the harmonic mean \( (p_1(1-x_1t)^{-1}+p_2(1-x_2t)^{-1}+\cdots+p_k(1-x_kt)^{-1})^{-1} \) into a power series in \( t \) of the form  \( \sum_{l\geq 0} q_l(x_1, . . . , x_k) t^l, \) the coefficient polynomials \( q_l =q_l(\underline{p},\underline{x}) \)  are nonpositive on the nonnegative orthant \( \mathbb R_{\geq 0}^k. \) We show that even under the more general hypothesis that \( \underline{p}=(p_1, . . . , p_k) \) is an arbitrary probability vector a stronger conclusion can be drawn: writing, say, \( x_i=h_i+\cdots +h_{k}, \) and expanding \( q_l \) in variables \( h_i \) results in a polynomial \( h_1, . . ., h_k \) with only negative coefficients.  The proof makes use of results in three earlier preprints.