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Let $p=(p_1,\ldots,p_n)$ be a given nondegenerate (i.e., all $p_i> 0$) probability distribution on $n$ points. Define the following function $$\Phi(b_1,\ldots,b_n)=\frac{\left(\sum_{k=1}^n b_k \sqrt{p_k}\right)^2}{\left(\sum_{k=1}^n b_k^2/k\right)}$$ whose domain is the simplex $\{(b_1,\ldots,b_k):\sum_{k=1}^n b_k=1,\quad b_k\geq 0,\quad k=1,\ldots, n\}.$

Claim 1: This function is convex-$\cap$ and attains a unique maximum on its domain.

I believe that an analytic solution to the maximization problem may be possible, but I am not sure if Lagrange multiplier methods are appropriate, not being an expert on optimization.

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