Inequality Problem - Didn't use 2015 to avoid Complicacy!

Algebra Level 5

\[\large{abc \left( a^{125} + b^{125} + c^{125} \right)^{16} \leq K \left( a^{2003} + b^{2003} + c^{2003} \right)}\]

Find the value of the smallest constant \(K\) such that, for any positive real numbers \(a,b,c\) , the above inequality satisfies. Submit the value of \(\ln(K)\) correct upto two decimal places as your answer.

Bonus: Can you find the smallest constant \(K\) in terms of \((p,q,n) \in \mathbb Z\) for the inequality below?

\[\large{\left(\prod_{i=1}^n x_i \right) \left(\sum_{i=1}^n x_i^p \right)^q \leq K \left(\sum_{i=1}^n x_i^{pq+n} \right)}\]


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