Upsetting inequality (à la Cauchy-Schwarz?)

Solution 1:

Consider some random variables $X$ and $Y$ such that, for every $i$, $(X,Y)=(n_iy_i,n_i(1-y_i))$ with probability $w_i$. The OP asks a proof of an inequality equivalent to $$ E(g(X,Y))\le g(E(X),E(Y)), $$ where, for every nonnegative $x$ and $y$, $$ g(x,y)=\sqrt{\frac{xy}{x+y+1}}. $$ The second partial derivatives $\partial^2_{xx}g$ and $\partial^2_{yy}g$ are negative and the determinant of the Hessian matrix of $g$ is $(xy+x+y)/(4xy(x+y+1)^3)$, which is positive. Hence both eigenvalues of the Hessian matrix are negative, the function $g$ is concave on its domain, and Jensen's inequality yields the result.