We are tasked with proving the inequality:

$a^a \cdot b^b \cdot c^c > abc^{\frac{a+b+c}{3}}$

Step-by-step solution:

Step 1: Restate the inequality

The inequality can be written as:

$a^a \cdot b^b \cdot c^c > abc^{\frac{a+b+c}{3}}$

Taking the natural logarithm of both sides, we aim to simplify the inequality. Applying the logarithmic property $\log(xy) = \log(x) + \log(y)$, we get:

$\log(a^a \cdot b^b \cdot c^c) > \log\left(abc^{\frac{a+b+c}{3}}\right)$

This simplifies to:

$a \log a + b \log b + c \log c > \log a + \log b + \log c + \frac{a+b+c}{3} \log c$

Step 2: Compare powers and simplify

To proceed,