Analytic function in open unit disc. How to prove that there exist an analytic function $f$ of $|z|<1$ onto itself such that $f(0)=1/2$, $f(1/2)=1/3$  and  $f(1/3)=1/4$.
 A: This is an example of the Pick-Nevanlinna interpolation problem. The existence of the function is equivalent to a positive semi-definiteness of a simple 3 by 3 matrix: 
http://en.wikipedia.org/wiki/Nevanlinna%E2%80%93Pick_interpolation
A: We could try transporting the problem to another domain that is easier to handle than the unit disk - I will use the standard map to the Poincaré half-plane $H = \{ z \in \mathbb C,\; \mathrm{Im} z > 0\}$ given by $h(w) =i\frac{1+w}{1-w}$. Your problem then becomes: to find a map $g: H \rightarrow H$ such that $g(i) = 3i$, $g(3i) = 2i$, $g(2i) = 5i/3$; or, to find a map $g: \mathbb C \rightarrow \mathbb C$ satisfying the same conditions plus $g(\mathbb R) \subset \mathbb R$.
Now, the condition $g(\mathbb R) \subset \mathbb R$ means that $g$ is a real-analytic map. Given that we have a finite number of points to interpolate, we will even decide that $g$ is a polynomial. To fix the imaginary axis, we will use a odd polynomial. The smallest odd polynomial interpolating the values we want is
$$ g(x) = (31x^5+415x^3+1464x)/360 \qquad \tiny\textrm{(bleuargh)}.$$
A solution to your original problem is therefore
$$ f = h^-1 \circ g \circ h, w \mapsto \frac{-360 w^5
 + 1613 w^4
 - 1917 w^3
 - 117 w^2
 + 713 w
 - 180}{-180 w^5
 + 713 w^4
 - 117 w^3
 - 1917 w^2
 + 1613 w
 - 360}.$$
EDIT:
(In the Poincaré half-plane, real polynomials are fine. In the unit disk, we could have proceeded directly in the same way with Fourier series.)
