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I'm reading "Lectures on Quasiconformal Mappings" by Lars Ahlfors. On page 16, while proving a lemma, he states two inequalities without justification. I would like to know why these inequalities hold.

book excerpt

The Cauchy-Schwarz inequality has been the key to several other inequalities so far, but I don't see how to apply that here. Thanks in advance for any help!

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I think the first inequality is supposed to be equality (and it is in my edition of the book), since

$$ \int\int_Q \rho^2 - 1 = \int\int_Q \rho^2 - \int\int_Q 1 = \int\int_{Q_1} \rho^2 + \int\int_{Q_2} \rho^2 - \int\int_Q 1 = |Q_1'| + |Q_2'| - |Q| = |Q'| - |Q| = 0, $$

where $|\cdot|$ denotes the measure of the set.

The second inequality is because $$ \int\int_Q \rho -1 = \int_0^1\int_0^m (\rho - 1) dx dy, $$

and $\int_0^m \rho dx$ gives the length of the image of a horizontal segment connecting the two vertical sides of $Q$, which will be at least $m$.

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