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

作者：Korhonen Risto^*

来源：Journal of Mathematical Analysis and Applications, 2009, 357(1): 244-253.

DOI：10.1016/j.jmaa.2009.04.011

• Picard's theorem
• Second main theorem
• Hyper-order
• Forward invariant
• Value distribution

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 is an element of 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 tau(z) = z alpha(n-1)z(1-1/n) ... alpha(1)z(1/n) alpha(0) with constant coefficients, then f circle tau equivalent to 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.

• 出版日期2009-9-1