I come here from a substantial application in statistics where I have reason to belive that the following ratio (function) is
$$f(X,Y)=\frac{1}{(2XY^2-X^2Y^2+X^2-2X+1)^{\frac{1}{2}}} \ge 1$$
for $X,Y \in [0,1]$. How should I go about showing this? Maybe somebody can brush up my maths.
Extension: the function above is a simplification for $Z=1$ in
$$g(X,Y,Z)=\frac{((1-X)Z^2+X)}{Z(2Z^2Y^2(X-X^2)+Z^2(X^2-2X+1)+Y^2X^2)^{\frac{1}{2}}} \ge 1$$
where now $X,Y,Z \in [0,1]$.
$$2XY^2-X^2 Y^2+X^2-2X+1 = 1-X(2-X)(1-Y^2)\leq 1.$$