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Nonlinear differential polynomials sharing a non-zero polynomial with finite weight

Abhijit Banerjee, Molla Basir AHAMED (2016)

Mathematica Bohemica

In the paper, dealing with a question of Lahiri (1999), we study the uniqueness of meromorphic functions in the case when two certain types of nonlinear differential polynomials, which are the derivatives of some typical linear expression, namely h n ( h - 1 ) m ( h = f , g ), share a non-zero polynomial with finite weight. The results obtained in the paper improve, extend, supplement and generalize some recent results due to Sahoo (2013), Li and Gao (2010). In particular, we have shown that under a suitable choice of...

Nonlinear differential polynomials sharing a small function

Abhijit Banerjee, Sonali Mukherjee (2008)

Archivum Mathematicum

Dealing with a question of Lahiri [6] we study the uniqueness problem of meromorphic functions concerning two nonlinear differential polynomials sharing a small function. Our results will not only improve and supplement the results of Lin-Yi [16], Lahiri Sarkar [12] but also improve and supplement a very recent result of the first author [1].

Non-maximal cyclic group actions on compact Riemann surfaces.

David Singerman, Paul Watson (1997)

Revista Matemática de la Universidad Complutense de Madrid

We say that a finite group G of automorphisms of a Riemann surface X is non-maximal in genus g if (i) G acts as a group of automorphisms of some compact Riemann surface Xg of genus g and (ii), for all such surfaces Xg , |Aut Xg| > |G|. In this paper we investigate the case where G is a cyclic group Cn of order n. If Cn acts on only finitely many surfaces of genus g, then we completely solve the problem of finding all such pairs (n,g).

Non-rectifiable limit sets of dimension one.

Christopher J. Bishop (2002)

Revista Matemática Iberoamericana

We construct quasiconformal deformations of convergence type Fuchsian groups such that the resulting limit set is a Jordan curve of Hausdorff dimension 1, but having tangents almost nowhere. It is known that no divergence type group has such a deformation. The main tools in this construction are (1) a characterization of tangent points in terms of Peter Jones' beta's, (2) a result of Stephen Semmes that gives a Carleson type condition on a Beltrami coefficient which implies rectifiability and (3)...

Non-recurrent meromorphic functions

Jacek Graczyk, Janina Kotus, Grzegorz Świątek (2004)

Fundamenta Mathematicae

We consider a transcendental meromorphic function f belonging to the class ℬ (with bounded set of singular values). We show that if the Julia set J(f) is the whole complex plane ℂ, and the closure of the postcritical set P(f) is contained in B(0,R) ∪ {∞} and is disjoint from the set Crit(f) of critical points, then every compact and forward invariant set is hyperbolic, provided that it is disjoint from Crit(f). It is further shown, under general additional hypotheses, that f admits no measurable...

Normal families and shared values of meromorphic functions

Mingliang Fang, Lawrence Zalcman (2003)

Annales Polonici Mathematici

Let ℱ be a family of meromorphic functions on a plane domain D, all of whose zeros are of multiplicity at least k ≥ 2. Let a, b, c, and d be complex numbers such that d ≠ b,0 and c ≠ a. If, for each f ∈ ℱ, f ( z ) = a ⇔ f ( k ) ( z ) = b , and f ( k ) ( z ) = d ⇒ f ( z ) = c , then ℱ is a normal family on D. The same result holds for k=1 so long as b≠(m+1)d, m=1,2,....

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