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How might we show that $\Big|{b^2+d^2-a^2-c^2+i2ab+i2cd\over a^2+b^2+c^2+d^2+2}\Big|\le 1$ if we are given that $ad-bc=1$ and $a,b,c,d$ are real?

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@WillieWong: thank you!!! –  daniel May 14 '12 at 11:31
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sorry for not posting it originally as an answer. I got lazy at first, but then decided to practice what I preach. –  Willie Wong May 14 '12 at 11:37

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If you write $z = (ia + b)$ and $w = (ic + d)$, the denominator is $z^2 + w^2$ and the numerator is $|z|^2 + |w|^2 + 2$. Then you can use the triangle inequality.

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