Malmquist's theorem

Remarks

A proof with Nevanlinna theory compares the growth of both sides: if $R$ had degree $d$ in $w$, then $T(r, R(z, w)) = d\,T(r, w) + O(\log r)$, while $T(r, w')$ is at most $2T(r, w)$ plus a small error. This bounds the degree, and a study of the poles of $w$ excludes nonpolynomial dependence on $w$. Example: $w' = 1 + w^2$ has the transcendental meromorphic solution $w = \tan z$, and its right side is a polynomial of degree $2$ in $w$, as the theorem requires. In contrast, the solutions of $w' = w^3$ are $w = \pm (c - 2z)^{-1/2}$ and $w = 0$; they have branch points, as the theorem predicts, so no transcendental meromorphic solution exists. The hypothesis that the solution is transcendental is needed: $w = z$ solves $w' = 1 + (w - z)^3$, whose right side has degree $3$ in $w$.

Going deeper

For a student

The theorem is the first-order prototype of the classification behind the Painlevé equations, where the same question is asked for second-order equations. It also has an analogue for difference equations $w(z + 1) = R(z, w(z))$, studied in Nevanlinna theory of difference equations: transcendental meromorphic solutions of finite order force strong restrictions on $R$. Results of this kind are called Malmquist-type theorems.