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Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane.

Question. Can one show that the unit circle lies in $$\lambda(\{x+iy: -1\leq x < 1 , y>1/2\})?$$

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You wrote $1\leq x <1$. I'm sure that's not what you meant. –  Rasmus Oct 21 '11 at 7:46
    
There are still some $1$s that should be $-1$s. –  Rasmus Oct 21 '11 at 11:22

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