Let $n$ be a growing integer parameter, and suppose that $X_1,\dotsc,X_n$ are independent Bernoulli random variables with the probabilities of success $p_i:={\mathsf P}(X_i=1)$. If $X=X_1+\dotsb+X_n$ then, trivially, ${\mathsf E}(\sqrt X)\le\sqrt{np}$, where $p=(p_1+\dotsb+p_n)/n$. When can one expect ${\mathsf E}(\sqrt X)=o(\sqrt{np})$ to hold, and what estimates can be proven in this direction? (I actually would like to have an explicit and reasonable improvement as compared to the trivial $\sqrt{np}$ bound.) I suspect normal and Poisson approximations might be useful  are they?
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Having carefully checked the things, I am not really interested in the case where $pn\to 0$; and, when $pn\gg 1$, the observations of camomille and Ori resolve the problem, leading to ${\mathsf E}(\sqrt X)\gg\sqrt{np}$. Thanks to all those who have replied! 

