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?



The expectation of $\sqrt{B(n,p)}$
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! 


