Given $0 < x_1 < x_2 < x_3 < x_4 < 1$, how can I show that the following inequality holds:
$$ \frac{1}{R(x_1, x_3)}+\frac{1}{R(x_2, x_4)}<\frac{1}{R(x_1, x_2)}+\frac{1}{R(x_3, x_4)} $$ where $R(x, y)=\log(1+S(x,y))$ and
$$ S(x,y)=\frac{-x-y+\sqrt{a x^2y+(x+y)^2}}{2x} $$ and $a\ge 1$ is a constant.