If $y_i = c x_i$ for a constant $c > 0$, then the new harmonic mean is $HM_y = c \cdot HM_x$.

If $y_i = x_i + c$ for a constant $c > 0$, then the new harmonic mean is $HM_y = HM_x + c$.

If $y_i = \frac{1}{x_i}$, then the arithmetic mean of the new dataset is the reciprocal of the harmonic mean of the original dataset: $\bar{y} = \frac{1}{HM_x}$.

If $y_i = \frac{1}{x_i}$, then the harmonic mean of the new dataset is the reciprocal of the arithmetic mean of the original dataset: $HM_y = \frac{1}{\bar{x}}$.