Let $w = \frac{s_2}{s_1}$, where $s_1 = \bar{x}$ (empirical mean), and $s_2 = \tilde{x}$ (geometric mean). Furthermore, Jensen's inequality states that $g(EV) \leq Eg(V)$ where $g$ is a convex function. Use this to prove that $0 < w \leq 1.$
So far I have been able to show that $$ \left[E\left(\tilde{X}^{1/n}\right)\right]^n \leq \left[E\left(\tilde{X}^{1/n}\right)\right]^n = E(\tilde{X})$$ by using Jensen's inequality, but not really sure how to progress further. Would appreciate any help.
Using Jensen's inequality, \begin{align} \ln(\bar{x})&=\ln\left(\sum_{i=1}^n \frac{x_i}{n}\right) \\ &\ge\sum_{i=1}^n \frac{\ln(x_i)}{n} \\ &=\ln\left(\prod_{i=1}^n x_i^{1/n}\right)=\ln(\tilde{x}). \end{align}