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assume f(z) is an entire function [analytical on the whole plane], that holds for every n (natural) ==> if |z|=n^2 then |f(z)|<13. prove that f(z) is a constant function.

If f is not constant, then $\displaystyle {\rm lim}_{|z|\rightarrow\infty}|f|=+\infty$. But $\displaystyle f\bigg|_{\{z:|z|=n^2\}}$ remains bounded, a contradiction.

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assume f(z) is a Meromorphic function [its finite singular points are only poles] on C, that holds for every z (complex) ==> f(z)=f(2f(z)). find all the functions f(z) [general form].

Consider the function near a pole: $\displaystyle f(z)=(z-a)^mg(z)+h(z)$, where $\displaystyle m<0$ and $\displaystyle h(a), g(a)$ are finite. Write $\displaystyle f=f(2f)$ and show that poles change order.

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ssume f(z) is periodical and analytical on a domain D that contains the point z=infinity. prove that f(z)=const on D.

Again, we must have $\displaystyle {\rm lim}_{|z|\rightarrow\infty}|f|=+\infty$. But f is periodical, and thus remains bounded.

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assume f(z) is analytical on a bounded domain D, and continuous on D* (with the "boundary"). assume also that f is not constant on D, but |f| is constant on the "boundary". prove that f has a zero on D.

Suppose not. Let $\displaystyle |f|\bigg|_{\partial D}=c$. Consider a fixed $\displaystyle a\in D^o$ and the function $\displaystyle g(z)=\frac{c(z-a)}{(c+1){\rm sup}_{w\in \partial D}|w-a|}$, which has a zero at z=a. Now $\displaystyle |g(z)|\bigg|_{z\in \partial D}\leq c/(c+1)<c= |f(z)|\bigg|_{z\in \partial D}$. Apply Rouche's theorem on the curve $\displaystyle \partial D$.

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prove that f(z) and h(z)=f(z+z^2) have the same "kind" of singular point on z=0, and if it is a zero or a pole, then the order is also the same.

Near zero, let $\displaystyle f(z)=z^mg(z)+z^kv(z), \ k<0$. Note that

$\displaystyle h(z)=(z+z^2)^mg(z+z^2)+(z+z^2)^kv(z+z^2)=z^mg(z+z^ 2)+z^kv(z+z^2)+w(z)$,

where w is analytic at 0.