lower bounding logarithm of sums
Solution 1:
Yes, it is true. Jensen's inequality was applied in the last step $$\log \left( \frac{1}{n} \sum \limits_{i=1}^{n}{\alpha_i} \right) \geq \frac{1}{n} \sum _{i=1}^n {\log(\alpha_i)},$$ since $\log$ is concave.