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Find all functions f which are meromorphic in a neighborhood of $\left\{|z|\le 1 \right\}$ and such that $|f(z)|=1$ for $|z|=1$ , f has a double pole at $z=\frac{1}{2}$, a triple zero at $z=-\frac{1}{3}$ and no other zeros or poles in $\left\{|z|<1 \right\}$

I have no idea how can I do this problem :S! Maybe using laurent expansion and uniqueness of that , but I don't know how can I find the general form of such functions, please help me :/ I want to see an example.

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up vote 1 down vote accepted

You may use idea of Blaschke product to construct desired function.

Function $f_0(z)=\left(\dfrac{z+\dfrac{1}{3}}{1+\dfrac{1}{3}z}\right)^3$ has zero of multiplicity $3$ at point $z_2=-\dfrac{1}{3}$. Analogously, function $f_p(z)=\left(\dfrac{z-\dfrac{1}{2}}{1-\dfrac{1}{2}z}\right)^2$ has zero of multiplicity $2$ at point $z_p=\dfrac{1}{2}$. Their quotient $g(z)=\dfrac{f_0(z)}{f_p(z)}$ will be meromorphic in the unit circle with predescribed poles and zeroes. At last, arbitrary rotations $f(z)=e^{i\varphi}g(z)$ will possess the same properties.

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well i'm very stupid... I don't know how to use such Blaschke products , I don't know nothing about that – Daniel Oct 15 '12 at 18:29
Oh I found something about Blaschke products, it's a characterization that will be helpful , thanks for the answer – Daniel Oct 15 '12 at 19:48

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