Let $S=\{(x_i, y_i)\}_{i=1...n} \in [0,1]^{2n}$ bet a tuple of ordered pairs, and let $A, H$ denote the arithmetic and harmonic mean. Then $$ \sup_S (H(\underset{i}{A}(x_i),\underset{i}{A}(y_i)) - \underset{i}{A}(H(x_i, y_i))) = \begin{cases} 0.5,n\text{ is even}\\\ 0.5 - \frac{1}{2n^2},\text{else} \end{cases} $$ We found a proof (Opitz and Burst, 2019: Macro F1 and Macro F1) by showing that the difference can be increased by intelligently swapping variables and then setting them to either 0 or 1. The proof is fairly long, and we are wondering: Is there a simple way to show this bound?

Let $S=\{(x_i, y_i)\}_{i=1...n} \in [0,1]^{2n}$ bet a tuple of ordered pairs, and let $A, H$ denote the arithmetic and harmonic mean. Then $$ \sup_S (H(\underset{i}{A}(x_i),\underset{i}{A}(y_i)) - \underset{i}{A}(H(x_i, y_i))) = \begin{cases} 0.5,n\text{ is even}\\ 0.5 - \frac{1}{2n^2},\text{else} \end{cases} $$ We found a proof (Opitz and Burst, 2019: Macro F1 and Macro F1) by showing that the difference can be increased by intelligently swapping variables and then setting them to either 0 or 1. The proof is fairly long, and we are wondering: Is there a simple way to show this bound?