I am trying to condition on the argument for the following two cases where $x_1,x_2,y_1,y_2\in R$,
Case 1.
$\begin{array}{l} \sum\limits_{i = 1}^2 {\sum\limits_{j = 1}^2 {Q\left( {{{\left( { - 1} \right)}^i}{x_1}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_1}} \right)} {{\log }_2}\left( {Q\left( {{{\left( { - 1} \right)}^i}{x_1}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_1}} \right)} \right)} \\ > \sum\limits_{i = 1}^2 {\sum\limits_{j = 1}^2 {Q\left( {{{\left( { - 1} \right)}^i}{x_2}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_2}} \right)} {{\log }_2}\left( {Q\left( {{{\left( { - 1} \right)}^i}{x_2}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_2}} \right)} \right)} \end{array}$
Where, $Q\left( x \right) = \frac{1}{{\sqrt {2\pi } }}\int_x^\infty {{e^{ - \frac{{{v^2}}}{2}}}dv} $. In this case, I guess if $x_1>x_2$ and $y_1>y_2$ the above relation holds true. But I don't know how to show that analytically.
Case 2.
$\begin{array}{l} \sum\limits_{i = 1}^2 {\sum\limits_{j = 1}^2 {Q\left( {{{\left( { - 1} \right)}^i}{x_1}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_1}} \right)} {{\log }_2}\left( {Q\left( {{{\left( { - 1} \right)}^i}{x_1}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_1}} \right)} \right)} \\ > \sum\limits_{i = 1}^2 {\sum\limits_{j = 1}^2 {Q\left( {{{\left( { - 1} \right)}^i}{x_2}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_2}} \right)} {{\log }_2}\left( {Q\left( {{{\left( { - 1} \right)}^i}{x_2}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_2}} \right)} \right)} \\ + \sum\limits_{i = 1}^2 {\sum\limits_{j = 1}^2 {Q\left( {{{\left( { - 1} \right)}^i}{x_3}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_3}} \right)} {{\log }_2}\left( {Q\left( {{{\left( { - 1} \right)}^i}{x_3}} \right)Q\left( {{{\left( { - 1} \right)}^j}{y_3}} \right)} \right)} \end{array}$
For this case I don't know how to condition on $x_1, x_2,x_3$, and $y_1,y_2,y_3$ such that I can show the above inequality (If any theorem exists for this kind of generic problem).
Any suggestion would be very much appreciated?