I once read some books about Nevanlinna theory, most of them will discuss the Nevanlinna main theorem small function theorem under some conditions. While, I know little motivation of small function theorem in Nevanlinna theory. And also I want to know elegant application of this theorem. I also have the same question about moving planes or hypersurface for holomorphic curve.
The reference and comments will be appreciated.
One motivation is mentioned in Yamanoi's paper where the second main theorem for small functions is proved: a theorem of Picard says that if we have meromorphic solutions $x(z),y(z)$ of $F(x,y)=0$, where $F$ is a polynomial then the genus of $F(x,y)=0$ is at most one. Then one wants a generalization to equations of the form $F(x(z),y(z),z)=0$, where $F$ is a polynomial in three variables, or, more generally, a polynomial in $x$ and $y$ with coefficients from some filed of functions of $z$. This equation in turn arises in analytic theory of differential equations where they consider equations $F(y',y,z)=0$. One can consider other functional equations of this kind, not only differential. (In general, one of the main applications of Nevanlinna theory is solving functional equations).
But there is a much deeper reason: look at the paper of MR3056292 Yamanoi, Katsutoshi, Zeros of higher derivatives of meromorphic functions in the complex plane. Proc. Lond. Math. Soc. (3) 106 (2013), no. 4, 703–780.
In this remarkable paper, a problem not involving "small functions" is solved. But the main tool is an improved version of the small (rational) function SMT.
Besides that, there is a feeling among the specialists, that "small functions" SMT is closer to number theory than the usual SMT, I mean the Vojta analogy between Nevanlinna theory and Diopnantine approximation. I mean roughly speaking that the usual Nevanlinna's SMT is heavily relying on the fact that the derivative of a constant is zero. In Vojta's analogy there is no counterpart of this fact. So if one tries to avoid using this fact in Nevanlinna theory, one comes closer to the arguments which can perhaps be extended to number theory. But this is just philosophy, I cannot make a precise statement.
