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...

Suppose that f is meromorphic in the plane and that F and G are given by

We shall prove theorems on nonexistence of certain types of vector fields on a compact manifold with a positive definite Riemannian metric whose Ricci curvature is either everywhere positive or everywhere negative. Actually we shall have some relaxations of the requirements both as to curvature and as to compactness. We shall deal with real spaces with a customary metric and with complex analyt...

Meromorphic functions with a given growth of spherical derivative on the complex plane are described in terms relative location a-points functions. The result obtained allows one to construct an example meromorphic function ? slow Nevanlinna characteristics and arbitrary derivative. In addition, based universality property Riemann zeta-function, we estimate ?(z).

is called Riemann’s Theorem and guarantees the existence of functions whose poles can only occur at certain places. In fact, if D = ∑ ni(Pi) − ∑ mj(Qj), then H(D) consists of all functions f with zeros of order ≥ mj at Qj and poles of order ≤ ni at Pi. Riemann’s original motivation was to construct meromorphic functions on compact Riemann surfaces. For example, the meromorphic functions on a to...

In this paper we introduce the concept of Abelian integrals in differential equations for an arbitrary vector bundle on P1 with a meromorphic connection. In this general context we give an upper bound for the numbers we are looking for. Let V be a locally free sheaf (vector bundle) of rank α on P and D = ∑r i=1 mici be a positive divisor in P , i.e. all ci’s are positive. We denote by C the set...

where F is a polynomial in the first k+ 1 variables, whose coefficients are analytic functions of the independent variable z. If the conditions of Cauchy's theorem for the existence and uniqueness of the solution are satisfied, then (0.1) determines an analytic function in a neighbourhood of a given point z0. One of the most difficult problems in the analytic theory of differential equations is...

the purpose of this paper is to derive various useful subordination properties and characteristics for certain subclass of multivalent meromorphic functions, which are defined here by the multiplier transformation. also, we obtained inclusion relationship for this subclass.