Given a sample space $\Omega=\{ 1,\cdots,N \}$, a random variable $x$ defined on $\Omega$ that takes value $x_1,\cdots,x_N$, and a set of strictly positive real numbers $w_1,\cdots,w_N$. Define for any probability distribution $\lbrace p_i \rbrace$ on $\Omega$ another probability distribution $\{q_i\}$ as $q_i(\{ p_i \})=w_i p_i /\sum_k w_k p_k$. The question is what is the maximum absolute difference of expectations of x under p and under q, that is,

$\max_p |\sum_i x_i p_i -\sum_i x_i \frac{w_i p_i}{\sum_k w_k p_k}|$

I guess that the maximum is

$\max_{i,j} |(x_i-x_j)\frac{\sqrt{w_i}-\sqrt{w_j}}{\sqrt{w_i}+\sqrt{w_j}}|$

However I don't have any proof to know whether this is correct. Is this correct? This seems to be a fairly straight-forward problem and most likely has a known result. Please let me know how this is solved.