Picard Type Theorem Research Articles

Regions of meromorphy and value distribution of geometrically converging rational functions

Let D be a region, { r n } n ∈ N a sequence of rational functions of degree at most n and let each r n have at most m poles in D, for m ∈ N fixed. We prove that if { r n } n ∈ N converges geometrically to a function f on some continuum S ⊂ D and if the number of zeros of r n in any compact subset of D is of growth o ( n ) as n → ∞ , then the sequence { r n } n ∈ N converges m 1 -almost uniformly to a meromorphic function in D. This result about meromorphic continuation is used to obtain Picard-type theorems for the value distribution of m 1 -maximally convergent rational functions, especially in Padé approximation and Chebyshev rational approximation.

Shared values, Picard values and normality

It is known that a family of meromorphic functions is normal if each function in the family shares a 3-element set with its derivative. In this paper we consider value distribution and normality problems with regard to 2-element shared sets. First we construct an example, by use of the Weierstrass doubly periodic functions, to show that a 3-element shared set can not be reduced to a 2-element shared set in general. We obtain a new criterion of normal families and new Picard-type theorems. The proofs make use of some results in complex dynamics. More examples are constructed to show that our assumptions are necessary.

Differential polynomials which share a value with their derivative

Abstract Let P be a differential polynomial with constant coefficients and F a family of meromorphic functions in the unit disk. We study the situation that P[f] and P´[f] share a finite nonzero value CM for all f∈F and give conditions on P for F to be normal. Furthermore, a corresponding Picard type theorem for entire functions is given. Connections to Picard type theorems for exceptional values of differential polynomials are shown. The general results are applied to several specific differential polynomials such as P[u]:=u n +au (k) or P[u]:=u n u (k).

Open discrete mappings having local ACL n inverses

We consider open, discrete mappings between domains from R n satisfying condition (N), having local ACL n inverses on D∖B f , so that μ n (B f ) = 0, H*(·, f) < ∞ on B f and . For this class of mappings (or even for larger classes of open, discrete mappings) we generalize the important modular inequality of Poleckii, also using the modular estimates of the spherical rings from Cristea (Local homeomorphism having local ACL n inverses, Complex Var. Elliptic Equ. 53(1) (2008), 77–99). We continue the work from the same paper by generalizing some basic facts from the theory of quasiregular mappings. We give equicontinuity results, Picard, Montel and Liouville type theorems, estimates of the modulus of continuity, analogues of Schwarz's lemma, eliminability results and boundary extensions theorems. Together with the multiple extensions of Zoric's theorem from the paper, we establish strong generalizations of one of the most important theorems from the theory of quasiregular mappings. We also extend similar results given in some recent classes of functions larger than the class of quasiregular functions, as the class of mappings of finite distortion and satisfying condition (𝒜), or the class of mappings of finite dilatation with dilatation in the bounded mean oscillation class.

On the Bloch Heuristic Principle, normality, and totally ramified values

Guided by the Bloch Heuristic Principle, we give a characterization of entire functions of order ≦ 1 with two totally ramified values, a normality criterion for analytic functions in plane domains, and a Picard type theorem. The results are complemented by various examples for completeness.

Big Picard's theorems for holomorphic mappings of several complex variables into [formula omitted] with moving hyperplanes

Motivated by the accomplishment of the second main theorem with moving targets, many authors studied the moving target problems in value distribution theory and related topics. In this paper, we prove some big Picard's theorems for holomorphic mappings of several complex variables into P N ( C ) with moving hyperplanes, related to Nochka's little Picard-type theorems. As its application, we give a new quasi-normal criterion for families of meromorphic mappings of several complex variables into P N ( C ) with moving hyperplanes. The new results in this paper greatly improve some earlier related results.

Normal families of meromorphic mappings of several complex variables into PN (C) for moving targets

Motivated by Ru and Stoll’s accomplishment of the second main theorem in higher dimension with moving targets, many authors studied the moving target problems in value distribution theory and related topics. But thereafter up to the present, all of researches about normality criteria for families of meromorphic mappings of several complex variables into PN (C) have been still restricted to the hyperplane case. In this paper, we prove some normality criteria for families of meromorphic mappings of several complex variables into PN(C) for moving hyperplanes, related to Nochka’s Picard-type theorems. The new normality criteria greatly extend earlier related results.

Zeros and interpolation by universal Taylor series on simply connected domains

By investigating the relation between growth and value distribution A. Melas established a qualitative version of a Picard type theorem for universal Taylor series, that is: every universal Taylor series on the open unit disk arbitrarily fast. Hence in view of Melas' work our result is the best possible. We also study the problem of interpolation by universal Taylor series.

Rigid analytic Picard theorems

We prove a geometric logarithmic derivative lemma for rigid analytic mappings to algebraic varieties in characteristic zero. We use the lemma to give a new and simpler proof (at least in characteristic zero) of Berkovich's little Picard theorem, which says there are no nonconstant rigid analytic maps from the affine line to nonsingular projective curves of positive genus, and of Cherry's result that there are no nonconstant rigid analytic maps from the affine line to Abelian varieties. Furthermore, we use the lemma to prove new theorems of little and big Picard type for dominant mappings, in close analogy with Griffiths and King. For the little Picard type theorem, we prove that if X is a smooth projective variety with a simple normal crossings divisor D such that ( X, D ) has nonnegative logarithmic Kodaira dimension, then there are no dominant rigid analytic maps f from A m to X \ D . For the big Picard type theorem, we prove that if Y is a nonsingular rigid analytic space, E is an effective simple normal crossings divisor on Y , and if X is a smooth projective variety with a simple normal crossings divisor D such that ( X, D ) is of log-general type, then any dominant rigid analytic map f : Y \ E →. X \ D extends to an analytic map from Y to X .

Hayman’s Alternative and Normal Families of Non-vanishing Meromorphic Functions

We show that if \(\cal F\) is a family of non-vanishing meromorphic functions in the unit disk D with P[f](z) ≠ 1 for all z ∈ D and all \(f \in {\cal F}\) where P is a differential polynomial satisfying certain conditions, then \(\cal F\); is normal. This generalizes former results of W. Schwick [19] and of M.-L. Fang [6]. Furthermore, we give the corresponding Picard type theorems generalizing Hayman’s Alternative.

On Meromorphically Normal Families of Meromorphic Mappings of Several Complex Variables into PN(C)

We shall prove some meromorphic normality criteria for families of meromorphic mappings of several complex variables into PN(C), the complex N-dimensional projective space, related to Nochka's Picard type theorems. Some related results (e.g., extension theorems, improved normality criteria, and quasi-normality criteria) will be obtained also. The technique in this paper mainly depends on Stoll's normality criteria for families of non-negative divisors on a domain of Cn.

Harnack inequality and hyperbolicity for subelliptic p -Laplacians with applications to Picard type theorems

Harnack inequality and hyperbolicity for subelliptic p -Laplacians with applications to Picard type theorems

Covering properties of derivatives of meromorphic functions

The Ahlfors theory of covering surfaces yields a striking generahzation of Picard's theorem. We discuss whether Picard type theorems where certain derivatives omit values admit similar generahzations.

Normality criteria for families of holomorphic mappings of several complex variables into 𝑃^{𝑁}(𝐶)

By applying the heuristic principle in several complex variables obtained by Aladro and Krantz, we shall prove some normality criteria for families of holomorphic mappings of several complex variables into P N ( C ) P^N(C) , the complex N-dimensional projective space, related to Green’s and Nochka’s Picard type theorems. The equivalence of normality to being uniformly Montel at a point will be obtained. Some examples will be given to complement our theory in this paper.

A Picard Type Theorem for Holomorphic Curves

Let P be complex projective space of dimension m, π : Cm+1\{0} → P the standard projection and M ⊂ P a closed subset (with respect to the usual topology of a real manifold of dimension 2m). A hypersurface in P is the projection of the set of zeros of a non-constant homogeneous form in m+ 1 variables. Let n be a positive integer. Consider a set of hypersurfaces {Hj} j=1 with the property M ∩ ⋂ j∈I Hj  = ∅ for every I ⊂ {1, . . . , 2n+ 1}, |I| = n+ 1. (1)

Circle packing immersions form regularly exhaustible surfaces

Circle packing imbeddings determine quasiconformai mappings which, in certain situations, approximate conformal mappings. In the more general case of circle packing immersions, it has not been proven that the associated immersion mappings are again quasiregular (i.e., of bounded dilatation but not necessarily univalent). If this were true, then their muitisheeted image surfaces would be regularly exhaustible and consequently Ahlfors' value distribution theory for covering surfaces could be applied to them. In this paper, although we do not settle the question or. whetner such immersions are quasiregular, we are able to prove directly from length-area estimates that the property of being regularly exhaustible holds for the muitisheeted image surfaces. Thus, for example, a Picard type theorem holds: The image of an entire circle packing immersion omits at most one finite value in the complex plane,

A Picard type theorem for quasiregular mappings of $\mathbb R^n$ into $n$-manifolds with many ends

A Picard type theorem for quasiregular mappings of $\mathbb R^n$ into $n$-manifolds with many ends

Sharing values and normality

According to Bloch's principle every condition, which reduces a meromorphic function in the plane to a constant, makes a family of meromorphic functions in a domain G normal. Although the principle is false in general ([6]), many authors proved normality criteria for families of meromorphic functions by starting from picard type theorems ([1], [7], [8]). Yang ([911 expanded such considerations to singular direcnons. It seems. however, that an attempt has not yet been made to prove normality criteria using conditions known from sharing value theorems (two meromorphic functions f and g share a e I~ iff f -1 ({a}) = 0 -1 ({a})). Nevanlinna's famous five point theorem says that two meromorphic functions in the plane which share five distinct values are identical. But if each pair f and g of meromorphic functions of a family shares five fixed values a t, the sets f-1 ({as}) are independent from f and the normality follows immediately from Montel's theorem. Therefore it is more interesting to discuss conditions, where a function f e F is not compared with all other functions in the family, but with specia! functions related to f. Mues and Steinmetz ([5]) proved

The Picard type theorem for essential singularities of solutions of systems of n rational differential equations

In the previous paper [4], for systems of n rational differential equations, we defined several concepts (e.g., the singular initial set Y, vertical singularity sets of the kth kind (k = 1, . . . . n + l), covering singularity sets of the kth kind (k= 1, . . . . n), the fixed singularity set of the kth kind Ok (k = 1, . . . . n + l), and so on), and proved the theorems which are generalizations of Painleve’s theorem. In this paper, using these concepts and the theorems, we study cluster sets of solutions of systems of n rational differential equations at essential singularities and prove the Picard type theorem for essential singularities of solutions. Through this paper, let DC C be a domain, &, the integral domain of holomorphic functions on D with the independent variable x, and hs-Yl 9 .--7 y,] the polynomial ring in y,, . . . . y, over Q,. As is mentioned in

Picard-type theorems for holomorphic mappings

where c. are constants and f' :C § T(W) is the derivative of the curve f. J Proof. We start the proof of the theorem by recalling some facts from algebraic geometry (see [2] for details). Let W be a compact complex manifold, D be a divisor on W with normal intersection, m = dim H~ be the dimension of the space of holomorphic differentials, and m + k = dim H ~ D)) be the dimension of the space of D-logarithmically meromorphic 1-forms on W.