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Given a function $$f(x,y)=\dfrac{(1-2ax)(1-2by)}{(1+2ax)(1+2by)}$$ for all constant $0\leq a\leq 1$, $0\leq b\leq 1$ and $x>0$, $y>0$.

Prove $\vert f(x,y)\vert\leq 1$.

\begin{align} \vert f(x,y)\vert&=\left\vert\dfrac{(1-2ax)(1-2by)}{(1+2ax)(1+2by)}\right\vert\\ &=\dfrac{\vert(1-2ax)\vert\vert(1-2by)\vert}{\vert(1+2ax)\vert\vert(1+2by)\vert}\\ \end{align}

Now, I don't know prove it. I don't know how to start the proof. What should I do to prove it?

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1 Answer 1

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Note that $|1-2ax|\leq|1|+|2ax|=1+2ax$ for $a,x>0$, and so $|1-2ax|/|1+2ax|\leq 1$.

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