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

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 n∈N and three distinct values of a meromorphic function f of hyper-order less than 1/n2 have forward invariant pre-images with respect to a fixed branch of the algebraic function τ(z)=z+αn−1z1−1/n+⋯+α1z1/n+α0 with constant coefficients, then f○τ≡f. 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.