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?