I am currently looking at different means and their properties. Proving that the Arithmetic Mean has the property from the question for nonnegative numbers is trivial, but I kinda struggle at seeing an argument why it is the case for the HM. Do you have an idea for that?
2026-04-02 18:37:47.1775155067
Why is the Harmonic Mean always bigger or equal than its minimum argument?
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Consider positive numbers $x_1, x_2, ..., x_n$ where $x_n$ is the least.
The harmonic mean is given by $$\frac{n}{1/x_1+1/x_2+...+1/x_n}\ge \frac{n}{1/x_n+1/x_n+...+1/x_n}=x_n.$$