Unicity of meromorphic functions and their derivatives

In this paper, we prove that if a transcendental meromorphic function f shares two distinct small functions CM with its kth derivative f^(k) (k>1), then f=f^(k). We also resolve the same question for the case k=1. These results generalize a result due to Frank and Weissenborn.     

Normal families of meromorphic functions concerning shared values

In this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family F of meromorphic functions in a domain D is normal if for each pair of functions f and g in F, f^′−afn and g^′−agn share a value b in D where n is a positive integer and a,b are two finite constants such that n?4 and a≠0. This result is not true when n?3.     

On the zeta-function of systems of nonlinear equations

We introduce the concept of zeta-function for a system of meromorphic functions f = (f ₁,..., f _n) in ?ⁿ. Using residue theory, we give an integral representation for the zeta-function which enables us to construct an analytic continuation of the zeta-function.     

Meromorphic functions sharing two small functions with its derivative

In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM^∗, and share b(z)IM^∗ with g(z)=a₁(z)f(z)+a₂(z)f^′(z). And a₁(z), a₂(z) and b(z) (a₂(z),b(z)?0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution.     

On exceptional values of entire and meromorphic functions

Letf(z) be meromorphic function of finite nonzero orderρ. Assuming certain growth estimates onf by comparing it withr ^ρ L(r) whereL(r) is a slowly changing function we have obtained the bounds for the zeros off(z) ?g (z) whereg (z) is a meromorphic function satisfyingT (r, g)=o {T(r, f)} asr → ∞. These bounds are satisfied but for some exceptional functions. Examples are given to show that such exceptional functions exist.     

Entire functions that share a small function with its derivative

In this paper, by studying the properties of meromorphic functions which have few zeros and poles, we find all the entire functions f(z) which share a small and finite order meromorphic function a(z) with its derivative, and f(n)(z)−a(z)=0 whenever f(z)−a(z)=0 (n?2). This result is a generalization of several previous results.     

Spherical derivatives and normal families

We show that a family of functions meromorphic in a plane domain D whose spherical derivatives are uniformly bounded away from zero is normal. Furthermore, we show that for each f meromorphic in the unit disk D, inf _z∈D f ^#(z) ≤ 1/2, where f ^# denotes the spherical derivative of f.     

Value sharing problems for differential and difference polynomials of meromorphic functions in a non-Archimedean field

In this paper we discuss the uniqueness problem for differential and difference polynomials of the form (f ^nm (z)f ^nd (qz + c))^(k) for meromorphic functions in a non-Archimedean field.     

On Value Distribution of Δ~nf

In this paper, we mainly study zeros and poles of the forward differences Δnf(z), where f(z) is a finite order meromorphic function with two Borel exceptional values.     

Picard values of <Emphasis Type="Italic">p</Emphasis>-adic meromorphic functions

We investigate Picard-Hayman behavior of derivatives of meromorphic functions on an algebraically closed field K, complete with respect to a non-trivial ultrametric absolute value. We present an analogue of the well-known Hayman’s alternative theorem both in K and in any open disk. Here the main hypothesis is based on the behaviour of |f|(r) when r tends to +∞ on properties of special values and quasi-exceptional values.We apply this study to give some sufficient conditions on meromorphic functions so that they satisfy Hayman’s conjectures for n = 1and for n = 2. Given a meromorphic transcendental function f, at least one of the two functions f′f and f′f ² assumes all non-zero values infinitely often. Further, we establish that if the sequence of residues at simple poles of a meromorphic transcendental function on K admits no infinite stationary subsequence, then either f′ + af ² has infinitely many zeros that are not zeros of f for every a ∈ K* or both f′ + bf ³ and f′ + bf ⁴ have infinitely many zeros that are not zeros of f for all b ∈ K*. Most of results have a similar version for unbounded meromorphic functions inside an open disk.     

Growth and poles of meromorphic solutions of some difference equations

This paper is devoted to studying the growth property and the pole distribution of meromorphic solutions f of some complex difference equations with all coefficients being rational functions or of growth S(r,f). We find the lower bound of the lower order, or the relation between lower order and the convergence exponent of poles of meromorphic solutions of such equations.     

Nevanlinna-Pick meromorphic interpolation: The degenerate case and minimal norm solutions

The Nevanlinna-Pick interpolation problem is studied in the class Sκ of meromorphic functions f with κ poles inside the unit disk D and with ‖f_‖L^∞(T)?1. In the indeterminate case, the parametrization of all solutions is given in terms of a family of linear fractional transformations with disjoint ranges. A necessary and sufficient condition for the problem being determinate is given in terms of the Pick matrix of the problem. The result is then applied to obtain necessary and sufficient conditions for the existence of a meromorphic function with a given pole multiplicity which satisfies Nevanlinna-Pick interpolation conditions and has the minimal possible L^∞-norm on the unit circle T.     

Value distribution and shared sets of differences of meromorphic functions

We investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions. In particular, we show that for a finite order transcendental meromorphic function f with λ(1/f)<ρ(f) and a non-zero complex constant c, if n?2, then fn(z)f(z+c) assumes every non-zero value a∈C infinitely often. This research also shows that there exist two sets S₁ with 9 (resp. 5) elements and S₂ with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, Ef(z)(Sj)=Ef(z+c)(Sj)(j=1,2) imply f(z)≡f(z+c). This gives an answer to a question of Gross concerning a finite order meromorphic function f and its shift.     

On the complex oscillation of meromorphic solutions of second order linear differential equations in the unit disc

The main purpose of this paper is to investigate the oscillation theory of meromorphic solutions of the second order linear differential equation f^″+A(z)f=0 for the case where A is meromorphic in the unit disc D={z:|z|<1}.     

Differential polynomials sharing one value

In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k+12. If fⁿ + af^(k) and gⁿ + ag^(k) share b CM and the b-points of fⁿ + af^(k) are not the zeros of f and g, then f and g are either equal or closely related.     

SOME NORMALITY CRITERIA OF MEROMORPHIC FUNCTIONS

Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.     

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.     

An extension of Picard's 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/n² have forward invariant pre-images with respect to a fixed branch of the algebraic function τ(z)=z+αn−1z1−1/n+?+α₁z1/n+α₀ with constant coefficients, then f○τ≡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.     

Normal family and the sequence of omitted functions

Let {fn} be a sequence of meromorphic functions on a plane domain D, whose zeros and poles have multiplicity at least 3. Let {hn} be a sequence of meromorphic functions on D, whose poles are multiple, such that {hn} converges locally uniformly in the spherical metric to a function h which is meromorphic and zero-free on D.If fn≠hn, then {fn} is normal on D.     

New results on applications of Nevanlinna methods to value sharing problems

Let $\mathbb{K}$ be a complete algebraically closed p-adic field of characteristic zero. We give a new Nevanlinna-type theorem that lets us obtain results of uniqueness for two meromrphic functions inside a disk, sharing 4 bounded functions CM. Let P be a polynomial of uniqueness for meromorphic functions in $\mathbb{K}$ or in an open disk, let f, g be two transcendental meromorphic functions in the whole field $\mathbb{K}$ or meromorphic functions in an open disk of $\mathbb{K}$ that are not quotients of bounded analytic functions and let α be a small meromorphic function with respect to f and g. We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove a result of uniqueness for functions: we show that if f′P′(f) and g′P′(g) share α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P′ satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k + 2 or n ≥ k + 3 used in previous papers by a new Hypothesis (G). Another consists of using the new Nevanlinna-type Theorem.