We make use of Jensen’s Inequality and the fact that \(log(x)\) is a concave function:

For concave f: \(f(\frac{\sum a_ix_i}{\sum a_i}) \geq \frac{\sum a_i f(x_i)}{\sum a_i}\)

\(f=log(X)\)

\(log(\frac{\sum a_ix_i}{\sum a_i}) \geq \frac{\sum a_i log(x_i)}{\sum a_i}\) \(\implies\) \(log(\frac{\sum a_ix_i}{\sum a_i}) \geq log(\prod (x_i)^{\frac{1}{a_i}})\)

Thus, \(\frac{\sum_i a_i x_i}{\sum_i a_i} \geq (\Pi x_i^{a_i})^{\frac{1}{\sum_i a_i}}\)

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