I have a complicated function:

\begin{equation} F(x,y) = \Big\lvert\cos(\sqrt{x^2 + 1}/2)\cos(\sqrt{y^2 + 1}/2) - \frac{xy + 1}{\sqrt{(x^2 + 1)(y^2+1)}} \sin(\sqrt{x^2 + 1}/2)\sin(\sqrt{y^2 + 1}/2) \Big\rvert, \end{equation} where $0 < x < 1$ and $x \geq y$. Numerically, I found out that $F$ is concave in $x$ and $y$. The way I tried to prove it is to show that $F(\lambda \vec{x} + (1-\lambda)\vec{y}) \geq \lambda F(\vec{x}) + (1-\lambda)F(\vec{y})$ where $\vec{x} = (x_1, x_2)$ and $\vec{y} = (y_1, y_2)$. However, it seems like the equation is too complicated to prove it in such way. Is there another simple way to prove it? Any advice would be appreciated.



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