Show $\prod_{i=1}^{n}x_i^{\alpha_i} \leq \sum_{i=1}^{n}\alpha_ix_i$ using convexity.

Question

Prove that $$\prod_{i=1}^{n}x_i^{\alpha_i} \leq \sum_{i=1}^{n}\alpha_ix_i$$ where $\alpha_i $ positive scalars with $\sum_{i=1}^n\alpha_i=1$ and $x_i$ are positive scalars.

I thought to express $\prod_{i=1}^{n}x_i^{\alpha_i} = \exp\left(\sum\alpha_i\log x_i\right)$ as the first step but cannot figure out what to do next. Your insight is appreciated.

Answer 1

Well, use Jensen's Inequality for the concave function $\ln$: $$\sum_{i=1}^n\,\alpha_i \,\ln\left(x_i\right)\leq \ln\left(\sum_{i=1}^n\,\alpha_i\,x_i\right)\,.$$