An extension of Picardʹs 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 / n 2 have forward invariant pre-images with respect to a fixed branch of the algebraic function τ ( z ) = z + α n − 1 z 1 − 1 / n + ⋯ + α 1 z 1 / n + α 0 with constant coefficients, then f ○ τ ≡ f . This is a generalization of Picardʹs 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.