Nonconstant Meromorphic Functions Research Articles

A unicity theorem for moving targets counting multiplicities

R. Nevanlinna showed, in 1926, that for two nonconstant meromorphic functions on the complex plane, if they have the same inverse images counting multiplicities for four distinct complex values, then they coincide up to a Möbius transformation, and if they have the same inverse images counting multiplicities for five distinct complex values, then they are identical. H. Fujimoto, in 1975, extended Nevanlinna’s result to nondegenerate holomorphic curves. This paper extends Fujimoto’s uniqueness theorem to the case of moving hyperplanes in pointwise general position.

Small functions and weighted sharing three values

This article studies the problem of the uniqueness of meromorphic functions that weighted sharing three values which improve some results given by Yi [Theorem 4, Yi, H.X., 1995, Unicity theorems for meromorphic functions that share three values. Kodai Mathematical Journal, 18, 300-314] and Ueda [Ueda, H., 1983, Unicity theorems for meromorphic or entire functions II. Kodai Mathematical Journal, 6, 26-36] and other authors. An application of these new results, if f and g are two distinct nonconstant meromorphic functions sharing 0, 1 and CM, and a is a nonconstant rational function, then N2)(r,1/(g-a))= [image omitted] An example shows that the latter result is not true for some transcendental small functions of f and g.

Meromorphic functions on compact Riemann surfaces and value sharing

Let X be a smooth and connected Riemann surface of genus g 0 and f: X P1, h: X P1 non-constant meromorphic functions on X. Fix an integer n 4 and assume the existence of n distinct points a1, , an P1 such that [image omitted] (set-theoretically) for every i. Here we prove that either f = h or [image omitted].

On meromorphic functions that share one value with their derivative

Abstract The following theorem has been proved by R.Brück. If a non-constant entire function ƒ and its derivative ƒ′ share the value 1 CM (counting multiplicities), and if , then ƒ′- 1 = c ( ƒ - 1 ) for some nonzero constant c . In this paper we shall prove that the above result is also true when ƒ is a non-constant meromorphic function.

Unicity Theorems for Meromorphic Functions and their Derivatives

Let f be a non-constant meromorphic function satisfying f f′ ≠ 0, and let a α ≢ 0 be a small function related to f. If f(z) = a(z) whenever f′(z) = a(z), then either f ≡ f′ or f(z) = 2a/(1 − ce−2z), where a, c are two non-zero constants and a(z) ≡ a.

Shared values of meromorphic functions on compact Riemann surfaces

Let S be a compact Riemann surface of genus g and gonality d. We derive upper bounds (in terms of g and/or d) for the number of values that two non-constant meromorphic functions on S can share. The case d = 2 (i.e., the surface is hyperelliptic or elliptic) is studied in more detail.

Fixed points of composite meromorphic functions and normal families

We show that there exists a function f, meromorphic in the plane C, such that the family of all functions g holomorphic in the unit disc D for which f ∘ g has no fixed point in D is not normal. This answers a question of Hinchliffe, who had shown that this family is normal if Ĉ\f(C) does not consist of exactly one point in D. We also investigate the normality of the family of all holomorphic functions g such that f(g(z)) ≠ h(z) for some non-constant meromorphic function h.

MEROMORPHIC FUNCTIONS THAT SHARE TWO SETS, II

This paper deals with the problem of uniqueness of meromorphic functions, and gets the following result: There exists a set S with 13 elements such that any two non-constant meromorphic functions f and g satisfying E¯(S, f) = E¯(S,g) and E¯({∞},f) = E¯({∞},g) must be identical. This is the best result on this question until now.

On differential equations satisfied by modular forms

We use the theory of modular functions to give a new proof of a result of P. F. Stiller, which asserts that, if t is a non-constant meromorphic modular function of weight 0 and F is a meromorphic modular form of weight k with respect to a discrete subgroup of SL 2 (ℝ) commensurable with SL 2 (ℤ), then F, as a function of t, satisfies a (k+1)-st order linear differential equation with algebraic functions of t as coefficients. Furthermore, we show that the Schwarzian differential equation for the modular function t can be extracted from any given (k+1)-st order linear differential equation of this type. One advantage of our approach is that every coefficient in the differential equations can be relatively easily determined.

Meromorphic functions sharing one finite set

Let f and g be two nonconstant meromorphic functions. If Θ(∞, f)+Θ(∞,g) > 3 2 , then there exists a set S with seven elements such that E( S, f) = E( S, g) implies f = g.

Uniqueness Theorems for Meromorphic Functions That Share Three Sets

In this article, we deal with the problem of uniqueness of meromorphic functions that share three sets, and obtain one set S with 5 elements such that any two nonconstant meromorphic functions ƒ and g satisfying E(S, ƒ) = E(S, g), E({0}, ƒ) = E({0}, g) and E({∞}, ƒ) = E({∞}, g) must be identical.

Inner Radius of Univalence for a Strongly Starlike Domain

The inner radius of univalence of a domain D with Poincare density ρD is the possible largest number σ such that the condition ∥ Sf ∥D = supw∈ D ρD (w)−2∥ Sf (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where Sf is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α.

Uniqueness problem of meromorphic mappings in several complex variables for moving targets

Nevanlinna showed that for two nonconstant meromorphic functions on the complex plane, if they have the same inverse images counting multiplicities for four distinct values, then they coincide up to a Möbius transformation, and if they have the same inverse images for five distinct values, then they coincide. Fujimoto and Smiley extended Nevanlinna's uniqueness theorems to the case of meromorphic mappings of several complex variables into the complex projective space for hyperplanes. Recently, Motivated by Ru Min and Stoll's accomplishment of the second main theorem for moving targets, Li Baoqin and Shirosaki proved some uniqueness theorems of entire functions in several complex variables and meromorphic functions in one complex variable, respectively, for moving targets. Using the techniques of value distribution theory in several complex variables, we prove some uniqueness theorems of meromorphic mappings of several complex variables into the complex projective space for moving targets.

Holomorphic vector bundles on compact complex non-algebraic surfaces

LetX be a smooth complex compact surface without non-constant meromorphic functions. Here we prove the existence of rank holomorphic vector bundles onX containing exactly one rank one saturated subsheaf.

UNIQUENESS POLYNOMIALS FOR COMPLEX MEROMORPHIC FUNCTIONS

A polynomial P(X) in [Formula: see text] is called a strong uniqueness polynomial for meromorphic functions if whenever there exist two non-constant meromorphic functions f and g and a complex non-zero constant c such that P(f) = cP(g), then we must have f = g. In this paper, we give a necessary and sufficient condition for a polynomial to be a strong uniqueness polynomial for meromorphic functions under the assumption that P(X) is injective on the roots of P′(X) = 0.

Uniqueness of meromorphic functions and differential polynomials

Using Nevanlinna value distribution theory, we study the uniqueness of meromorphic functions concerning differential polynomials, and prove the following theorem. Let f( z) and g( z) be two nonconstant meromorphic functions, n(≥ 13) be a positive integer. If f n ( f − 1) 2 f′ and g n ( g − 1) 2 g′ share 1 CM, then f ≡ g.

Meromorphic Functions That Share Fixed-Points

In this paper, we prove the following result: Let f(z) and g(z) be two nonconstant meromorphic(entire) functions, n≥11(n≥6) a positive integer. If fn(z)f′(z) and gn(z)g′(z) have the same fixed-points, then either f(z)=c1ecz2,g(z)=c2e−cz2, where c1,c2, and c are three constants satisfying 4(c1c2)n+1c2=−1, or f(z)≡tg(z) for a constant t such that tn+1=1.

Finite Dimensional de Branges Spaces on Riemann Surfaces

We study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real Riemann surface; these spaces are analogues of the spaces introduced by L. de Branges when the Riemann sphere is replaced by a compact real Riemann surface of a higher genus. In de Branges theory an important role is played by resolvent-like difference quotient operators Rα; here we introduce generalized difference quotient operators Ryα for any non-constant meromorphic function y on the Riemann surface. The spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators Ry1αi and Ry2α2, where y1 and y2 generate the field of meromorphic functions on the Riemann surface, which satisfy a supplementary identity, analogous to the de Branges identity for difference quotients. Just as the classical de Branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint (or nonunitary) operator, the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks.

Meromorphic Functions That Share Four Small Functions

In this paper, we prove that if two nonconstant meromorphic functions f and g share two small functions CM* and share other two small functions IM*, then f must be a quasi-Möbius transformation of g. This result is a generalization of several results obtained by G. G. Gundersen and Li-Yang.

Meromorphic solutions of

Hayman showed that for n≥9, there do not exist three nonconstant meromorphic functions f g, and h that satisfy . There are examples which show that this theorem does not hold when n = 6 and 1 ≥n≥4. In this paper we give an example which shows that this theorem does not hold when n=5.