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- Jun 20, 2014

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Here is this week's POTW:

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Suppose $a$ is a fixed complex number in the open unit disk $\Bbb D$. Consider the holomorphic mapping $\phi : \Bbb D \to \Bbb D$ given by $\phi(z) := (z - a)/(1 - \bar{a}z)$. Find, with proof, the average value of $\left\lvert\frac{d\phi}{dz}\right\rvert^2$ over $\Bbb D$, i.e., the integral $$\frac{1}{\pi}\iint_{\Bbb D} |\phi'(x + yi)|^2\, dx\, dy$$

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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Suppose $a$ is a fixed complex number in the open unit disk $\Bbb D$. Consider the holomorphic mapping $\phi : \Bbb D \to \Bbb D$ given by $\phi(z) := (z - a)/(1 - \bar{a}z)$. Find, with proof, the average value of $\left\lvert\frac{d\phi}{dz}\right\rvert^2$ over $\Bbb D$, i.e., the integral $$\frac{1}{\pi}\iint_{\Bbb D} |\phi'(x + yi)|^2\, dx\, dy$$

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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