The purpose of this article is to investigate the uniqueness of meromorphic functions sharing five small functions on annuli.

In this paper we introduce the notion of $L^{* }$-proximate order of meromorphic function and prove its existence.

This is a summary report of my recent articles. Nevanlinna theory asserts that each meromorphic mapping f of C into P(C) has few defects. However, it seems that meromorphic mappings with defects are very few. In this report, we shall show that for any given transcendental meromorphic mapping of C into P(C); there is a small deformation of f which has no Nevanlinna deficient hyperplanes in Pn(C)...

Let f be a non-constant meromorphic function, n, k be two positive integers and a(z)( 6≡ 0,∞) be a meromorphic small function of f . Suppose that f − a and (f)−a share the value 0 CM. If either (1) n ≥ k+1 and N(r,∞; f) = S(r, f), or (2) n > k + 1 and N(r,∞; f) = λ T (r, f)(λ ∈ [0, 1)), then f ≡ (f) and f assume the form f(z) = ce λ n , where c is a nonzero constant and λ = 1. This result shows...

Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ρ, then every asymptotic value of f , except at most 2ρ of them, is a limit point of critical values of f . We give several applications of this theorem. For example we prove that if f is a transcendental meromorphic function then f ′fn with n ≥ 1 takes every...

Let w(z) be a finite-order meromorphic solution of the second-order difference equation w(z + 1) + w(z − 1) = R(z, w(z)) (†) where R(z, w(z)) is rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else equation (†) can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painleve eq...

For a germ of a meromorphic function f = P Q , we offer notions of the mono-dromy operators at zero and at infinity. If the holomorphic functions P and Q are non-degenerated with respect to their Newton diagrams, we give an analogue of the formula of Varchenko for the zeta-functions of these monodromy operators. A polynomial f of (n + 1) complex variables of degree d determines a meromorphic fu...

is the Nevanlinna characteristic of f [13]. Meromorphic functions of finite order have been extensively studied and they have numerous applications in pure and applied mathematics, e.g. in linear differential equations. In many applications a major role is played by the logarithmic derivative of meromorphic functions and we need to obtain sharp estimates for the logarithmic derivative as we app...

Analogies between the Nevanlinna theory and the theory of heights in number theory have motivated to the determination the precise error term of the second fundamental theorem of the Nevanlinna theory. The Nevanlinna secondary deeciency, introduced by P.M. Wong, gives the error term in a certain sense. We rst prove that, from a topological point of view, almost all meromorphic functions have se...

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the ...