I'm trying to understand the difference between F-measure and G-mean. I know they are harmonic mean and geometric mean but this didn't explain the relationship to the classifier's performance
The three Pythagorean means (arithmetic, geometric, and harmonic) are special cases of the generalized mean. All mean functions have the property that if all your values are the same, the mean is the same value. The arithmetic mean has the property that if you decrease one value (e.g., precision) by the same amount that you increase another value (e.g., recall), the mean stays the same even though there are now unequal values. In contrast, the geometric and harmonic means both penalise any inequalities in the values. This article explains and visualizes it well.
For example, when $P = 0.9, R = 0.1$, the arithmetic mean is 0.5, the geometric mean is 0.3, and the harmonic mean is 0.18. The geometric mean and harmonic mean are more "pessimistic", and it is true in general that the arithmetic mean is equal or greater to the geometric mean, which is equal or greater to the harmonic mean.
Since the harmonic mean is more pessimistic than the geometric mean, using the geometric mean to evaluate a classifier's performance will favor precision and recall values that are closer together, compared to using the harmonic mean.