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 have forward invariant pre-images with respect ...

where f is meromorphic and g is entire (g may be meromorphic when f is rational). A nonlinear meromorphic function F (z) is called prime (pseudo prime) if every factorization of form (1) implies that either f is bilinear or g is linear (either f is rational or g is a polynomial). Clearly, a prime function is an analogy of a prime number. Over the past thirty years, many classes of prime or pseu...

A bstract We show that the differential equation for three-loop equal-mass banana integral can be cast into an ? -factorised form with entries constructed from (meromorphic) modular forms and one special function, which given as iterated of meromorphic forms. The allows a systematic solution to any order in dimensional regularisation parameter . alphabet integrals contains six letters.

This paper is related to notions adapted from fuzzy set theory the field of complex analysis, namely differential subordinations. work aims present new subordinations for which best dominant and subordinate are given, respectively. The original theorems proved in generate interesting corollaries particular choices functions acting as dominant. Here, this article, subordination results obtained ...