I have a set of real number bounded by 1 and $c$ (inclusive), where $c$ is greater than 1.
With $H$ the harmonic mean and $G$ the geometric mean of this set, I know that $1 \leq H \leq G \leq c$.
Does such a relationship, that would make use of $\log G$ (i.e. arithmetic mean of the log-values) rather than $G$, exist?
In particular, I am interested in the signs of $$1- \frac Hc + \log\frac Gc $$ and $$1 - H + \log G.$$
It is obvious that $1-\frac Hc$ is positive and $1-H$ is negative, and that $\log\frac Gc$ is negative and $\log G$ is positive. However, I cannot work out results for the sums.