I was trying to solve the following problem. But I dont know how to proceed. I would be really grateful if anybody would point me in the right direction.
Let $P = (p_1,p_2, \cdots, p_n)$ be a probability vector (That is $\sum p_i = 1$ and $p_i \geq 0$). Let $Q = (q_1,q_2, \cdots, q_n)$ be a permutation of the vector $P$.
If $I = \frac14 \left(\sum p_i \log \frac{p_i}{q_i}\right) + \frac14 \left(\sum q_i \log \frac{q_i}{p_i}\right)$ (involves the Kullback - Leibler divergence)
And $Z = \sum \sqrt{p_iq_i}$ (called the Bhattacharyya distance)
Prove that $I^2 + Z^2 \leq 1$
Thank you