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Let $\underline{b}=\{b_1,\dots,b_n\}$ be a fixed sequence of positive numbers, and let $a>0$ be a parameter.

Define $$ D(a;\underline{b}):=\frac{1}{\frac{1}{na}+\frac{1}{\sum_{i=1}^n b_i}} -\sum_{i=1}^n \frac{1}{\frac{1}{a}+\frac{1}{b_i}}. $$ By Jensen's inequality applied to the function $f_a(x)=\frac{1}{\frac{1}{a}+\frac{1}{x}}$ we get immediately that $D(a;\underline{b})\geqslant 0$, having equality if and only if $b_1=\dots=b_n=const$. Assume this is not the case, i.e. strict inequality holds.

Numerical experiments seem to suggest that the function $D(a;\underline{b})$ has a single maximum in $a\in(0,+\infty)$.

Can one (dis)prove this? Can one obtain an expression for the maximal value?

On a side note, using Holder's Defect formula we get the bounds $$ \frac{n Var(\underline{b})a^2}{2(a+\max(\underline{b}))^3} \leqslant D(a;\underline{b}) \leqslant \frac{n Var(\underline{b})a^2}{2(a+\min(\underline{b}))^3}. $$

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