Nevanlinna's Theorem Research Articles

On meromorphic solutions of nonlinear delay-differential equations

Using Cartan's second main theorem and Nevanlinna's theorem concerning a group of meromorphic functions, we obtain the growth and zero distribution of meromorphic solutions of the nonlinear delay-differential equation fn(z)+P(z)f(k)(z+η)=H0(z)+H1(z)eω1zq+⋯+Hm(z)eωmzq, where n,k,q,m are positive integers, η,ω1,⋯,ωm are complex numbers with ω1⋯ωm≠0, and P,H0,H1,⋯,Hm are entire functions of order less than q with PH1⋯Hm≢0. Especially for η=0, some sufficient conditions are given to guarantee the above equation has no meromorphic solutions of few poles.

Propagation sets of holomorphic curves

We consider a problem of whether a property of holomorphic curves on a subset [Formula: see text] of the complex plane can be extended to the whole complex plane. In this paper, the property we consider is the uniqueness of holomorphic curves. We introduce the propagation set. Simply speaking, [Formula: see text] is a propagation set if linear relation of holomorphic curves on the part of preimage of hyperplanes contained in [Formula: see text] can be extended to the whole complex plane. If the holomorphic curves are of infinite order, we prove the existence of a propagation set which is the union of a sequence of disks. (In fact, the method applies to the case of finite order.) For a general case, the union of a sequence of annuli will be a propagation set. The classic five-value theorem and four-value theorem of Nevanlinna are established in such propagation sets.

A short note on a pair of meromorphic functions in a p-adic field, sharing a few small ones

A new Nevanlinna theorem on q p-adic small functions is given. Let f, g, be two meromorphic functions on a complete ultrametric algebraically closed field $${\mathbb{K}}$$ of characteristic 0, or two meromorphic functions in an open disk of $${\mathbb{K}}$$ , that are not quotients of bounded analytic functions by polynomials. If f and g share 7 small meromorphic functions I.M., then $$f=g$$ . Better results hold when f and g satisfy some property of growth. Particularly, if f and g have finitely many poles or finitely many zeros and share 3 small meromorphic functions I.M., then $$f=g$$ .

On deviations, small functions and strong asymptotic functions of meromorphic functions

The paper addresses two long-standing problems: of extending the second main theorem of Nevanlinna to the case of small functions, and of finding an upper limit for the number of asymptotic functions of a function of finite lower order. Upper estimates of the sum of deviations and the numbers of strong asymptotic functions and strong functional asymptotic spots of meromorphic functions of finite lower order are presented. The structure of the set of Petrenko's deviations from small functions for meromorphic functions of finite lower order is examined. An analogue of Denjoy's question for strong asymptotic small functions of meromorphic functions of finite lower order is also considered.

Nevanlinna representations in several variables

Abstract We generalize to several variables the classical theorem of Nevanlinna that characterizes the Cauchy transforms of positive measures on the real line. We show that for the Loewner class, a large class of analytic functions that have non-negative imaginary part on the upper polyhalf-plane, there are representation formulae in terms of densely-defined self-adjoint operators on a Hilbert space. We find four different representation formulae and we show that every function in the Loewner class has one of the four representations, corresponding precisely to four different growth conditions at infinity.

Complex and $p$-Adic Meromorphic Functions $f'P'(f)$, $g'P'(g)$ Sharing a Small Function

Let K be a complete algebraically closed p-adic field of characteristic zero. We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk. Let f, g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions. We show that if f P (f) and g P (g) share a small function 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 Hypothesis (G). In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with (q − 1) times the characteristic function of the considered meromorphic function. Notation and definition: Let K be an algebraically closed field of characteristic zero, complete with respect to an ultrametric absolute value |. |. We will denote by E a field that is either K or C. Throughout the paper we denote by a a point in K. Given R ∈]0, +∞[ we define disks d(a, R) = {x ∈ K | |x − a| ≤ R} and disks d(a, R −) = {x ∈ K | |x − a| < R}. A polynomial Q(X) ∈ E[X] is called a polynomial of uniqueness for a family of functions F defined in a subset of E if Q(f) = Q(g) implies f = g. The definition of polynomials of uniqueness was introduced in [19] by P. Li and C. C. Yang and was studied in many papers [11], [13], [20] for complex functions and in [1], [2], [9], [10], [17], [18], for p-adic functions. Throughout the paper we will denote by P (X) a polynomial in E[X] such that P (X) is of the form b l i=1 (X − a i) ki with l ≥ 2 and k 1 ≥ 2. The polynomial P will be said to satisfy Hypothesis (G) if P (a i) + P (a j) = 0 ∀i = j. We will improve the main theorems obtained in [5] and [6] with the help of a new hypothesis denoted by Hypothesis (G) and by thorougly examining the situation with p-adic and complex analytic and meromorphic functions in order to avoid a lot of exclusions. Moreover, we will prove a new theorem completing the 2nd Main Theorem for p-adic meromorphic functions. Thanks to this new theorem we will give more precisions in results on value-sharing problems. 0 2000 Mathematics Subject Classification: 12J25; 30D35; 30G06. 0

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.

Meromorphic functions that weighted sharing three values and one pair

G. Brosch improved the theorem of Nevanlinna for four values theorem and proved that let f and g be two nonconstant meromorphic functions sharing 0, 1, ∞ CM, and let a and b be two finite complex numbers such that $a,b\not\in \{0,1\}$. If $f=a\Leftrightarrow g=b$, then f is a fractional linear transformation of g. In this paper we extend this theorem by using the idea of weighted sharing.

Sommation effective d’une somme de Borel par séries de factorielles

In this article, we consider the effective resummation of a Borel sum by its associated factorial series expansion. Our approach provides concrete estimates for the remainder term when truncating this factorial series. We then generalize a theorem of Nevanlinna which gives us the natural framework to extend the factorial series method for Borel-resummable fractional power series expansions.

A Nevanlinna theorem for superharmonic functions on Dirichlet regular Greenian sets

A generalization of Nevanlinna’s First Fundamental Theorem to superharmonic functions on Green balls is proved. This enables us to generalize many other theorems, on the behaviour of mean values of superharmonic functions over Green spheres, on the Hausdorff measures of certain sets, on the Riesz measures of superharmonic functions, and on differences of positive superharmonic functions.

URS AND URSIMS FOR P-ADIC MEROMORPHIC FUNCTIONS INSIDE A DISC

AbstractLet $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the $p$-adic main Nevanlinna Theorem holds for meromorphic functions inside an ‘open’ disc in $K$. Let $P_{n,c}$ be the Frank–Reinders’s polynomial$$ (n-1)(n-2)X^n-2n(n-2)X^{n-1}+ n(n-1)X^{n-2}-c\qq (c\neq0,\ c\neq1,\ c\neq2) $$and let $S_{n,c}$ be the set of its $n$ distinct zeros. For every $n\geq 7$, we show that $S_{n,c}$ is an $n$-points unique range set (counting multiplicities) for unbounded analytic functions inside an ‘open disc’, and for every $n\geq10$, we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of $17$ points. A better result is obtained for an analytic or a meromorphic function $f$ when its derivative is ‘small’ comparatively to $f$. In particular, for every $n\geq5$ we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the $p$-adic Nevanlinna Theory, and Frank–Reinders’s and Fujimoto’s methods used for meromorphic functions in $\mathbb{C}$. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.AMS 2000 Mathematics subject classification: Primary 12H25. Secondary 12J25; 46S10

The Nevanlinna Theorem of the Classical Theory of Moments Revisited

The canonical solutions of the truncated Hamburger moment problem (both in the classical and degenerate cases) are found. The Nevanlinna theorem which provides the noncanonical solutions of the truncated Hamburger problem is also rederived in the framework of the operator approach.

An Extension of the Corollary to Steinmetzs Theorem

In two recent papers, "A Simpler Proof of a Theorem of Steinmetz" (1989, J. Math. Anal. Appl.143, 290-294) and "An Extension of a Theorem of Steinmetz" (1991, J. Math. Anal. Appl.156, 287-292), the present authors gave a new proof of Steinmetz′s theorem that avoided use of the Second Fundamental Theorem of Nevanlinna. It was also shown in the second paper how to generalize Steinmetz′s theorem to handle the case where the inner function is not precisely the same in all cases. In this paper we shall extend these results even further.

The berry phase for filtering measurements, coherent states and the transition probabilities

The geometric phase of a cyclic filtering measurement is shown to be related to the shape invariant of geodesic projective triangles and it satisfies an inequality with respect to the transition probabilities. Holomorphic curves in coherent-state Grassmannians are considered. The geometric phase contained in the non-integrated Nevanlinna theorem is connected to the transition probabilities.

A fundamental inequality for meromorphic functions in an angular domain and its application

This paper is to study the characteristic functions of meromorphic functions for an angular domain. We show an inequality which is similar to the inequality in the second fundamental theorem of Nevanlinna, and estimate its remainder. Finally, as an application we obtain a sufficient and necessary condition of Borel directions.

Some generalizations of the unicity theorem of Nevanlinna

Some generalizations of the unicity theorem of Nevanlinna

Novel summability methods generalizing the Borel method

The paper deals with moment constant summability methods. A method is constructed which provides an analytic continuation of a function regular at the origin onto its Mittag-Leffler (principal) star. In this sense the method is optimal in contrast to the Borel one. In the next a one parameter family of such methods is introduced. The methods may be useful both in field theory and in statistical physics. Applications to the Nevanlinna theorem, the Rayleigh-Schro-dinger perturbation theory and the dispersion-like integral are given. The proofs of theorems can be easily adapted to the study of the Mellin transform of some entire functions and a simpler proof of asymptotic properties of the gamma function can be obtained.

On the second fundamental theorem of Nevanlinna

It is shown that a condition on the size of the exceptional set in the second fundamental theorem of Nevanlinna cannot be improved. The method is based on a construction of Hayman and also makes use of a quantitative version of a result of F. Nevanlinna about the growth of the characteristic function of a meromorophic function omitting a finite number of points

Picardsche ausnahmewerte von ableitungen gewisser meromorpher funktionen

It is an easy consequence of the second fundamental theorem of Nevanlinna that each derivative , of a transcendental meromorphic function g has at most one finite Picard value. Generalizing a result of Mues we prove that for g=fn+2 only zero can be this Picard value. The functions f=R exp P, R rationalP polynomial, show that Picard value zero can actually occur. We show that for kn, n3, (fk)(n) has Picard value zero only if f is of this form.

Degeneracy Theorems for Holomorphic Mappings Between Algebraic Varieties

Degeneracy theorems are proved for holomorphic mappings from affine algebraic manifolds to projective algebraic manifolds of equal dimensions. A mapping is degenerate if it satisfies a growth estimate and omits a set of $\kappa$-plane sections of positive capacity; the capacity being defined in terms of a singular integral. The capacity is a more delicate method of measuring the size of a set of $\kappa$-plane sections than Hausdorff measure and arises naturally by considering the singular integrals in the First Main Theorem of Nevanlinna.