An extension of Picard's theorem for meromorphic functions of small hyper-order

TL;DR

This paper extends Picard's theorem for meromorphic functions of small hyper-order by establishing a new second main theorem involving a composition operator, leading to a generalized Picard's theorem for such functions.

Contribution

It introduces a version of the second main theorem with a composition operator, generalizing Picard's theorem for meromorphic functions of small hyper-order.

Findings

Proves a second main theorem with a composition operator for small hyper-order functions.

Shows that functions with certain invariant pre-images satisfy a functional equation.

Generalizes Picard's theorem to a broader class of meromorphic functions.

Abstract

A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result it is shown that if three distinct values of a meromorphic function f of hyper-order less than 1/n^2 have forward invariant pre-images with respect to a fixed branch of the algebraic function t(z)=z+a_{n-1} z^{1-1/n}+...+a_1 z^{1/n}+a_0 with constant coefficients, then f(t(z)) = f(z) for all z. This is a generalization of Picard's theorem for meromorphic functions of small hyper-order, since the (empty) pre-images of the usual Picard exceptional values are special cases of forward invariant pre-images.