Inequality between geometric mean and harmonic mean

Question

Suppose that $x_1,\dots,x_n>0$. Prove that $$\sqrt[n]{x_1x_2\dots x_n}\geq \dfrac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}}$$ by using Jensen's inequality for some suitable function $f$.

Answer 1

Let $x_i=\frac{1}{a_i}.$

Thus, we need to prove that $$\ln\frac{\sum\limits_{i=1}^na_i}{n}\geq\frac{\sum\limits_{i=1}^n\ln{a_i}}{n},$$ which is true by Jensen for the concave function $\ln$.