On a new type of singular point for meromorphic functions in the open unit disk

In this note, we will confirm the existence of a new type of singular point of an admissible meromorphic function in the unit disk. This work is a counterpart of the result of a meromorphic function with order in the complex plane studied by W. C. Lin and S. Mori.     

On the unique range set of meromorphic functions

This paper studies the unique range set of meromorphic functions and shows that there exists a finite set such that for any two nonconstant meromorphic functions and the condition implies . As a special case this also answers an open question posed by Gross (1977) about entire functions and improves some results obtained recently by Yi.

     

T- and Hayman T-points of meromorphic functions for small functions in the unit disk

In this article, we consider the singular points of meromorphic functions in the unit disk. We prove the second fundamental theorem for the Ahlfors-Shimizu's characteristic in the unit disk in terms of Nevanlinna theory in the angular domains, and obtain the existence of T-points and Hayman T-points dealing with small functions as target.     

On a certain classes of meromorphic functions with positive coefficients

In this paper certain classes of meromorphic functions in punctured unit disk are defined. Some properties including coefficient inequalities, convolution and other results are investigated.     

On analytic functions in the unit disk with univalent derivatives

We establish conditions on an increasing sequence (n_p) of nonnegative integers that guarantee that a functionf analytic in the unit disk {z ¦z¦< 1} can be analytically continued to a disk of larger radius provided all of its derivatives are univalent in the disk.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 58–65.     

Uniqueness of admissible meromorphic functions in the unit disc

The uniqueness of admissible meromorphic functions in the unit disc is studied. Project supported by the National Natural Science Foundation of China (Grant No. 19601021).     

The size of the Julia set of meromorphic functions

We give a lower bound of the hyperbolic and the Hausdorff dimension of the Julia set of meromorphic functions of finite order under very general conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the behaviour of the successive derivatives of meromorphic functions on the final set

We study the behaviour of the sequence of successive derivatives of meromorphic functions at points of the so-called final set. We prove that, whereas in many cases this sequence tends to ∞, for a special class of meromorphic functions, it may have extremely wild behaviour. We also prove a connection between the derivatives of meromorphic functions from this class and so-called Dirichlet sets.     

On permutable meromorphic functions

We study the class \({\mathcal{M}}\) of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in \({\mathcal{M}}\), with at least one essential singularity, permutes with a non-constant rational map g, then g is a Möbius map that is not conjugate to an irrational rotation. For a given function \({f \in\mathcal{M}}\) which is not a Möbius map, we show that the set of functions in \({\mathcal{M}}\) that permute with f is countably infinite. Finally, we show that there exist transcendental meromorphic functions \({f : \mathbb{C} \to \mathbb{C}}\) such that, among functions meromorphic in the plane, f permutes only with itself and with the identity map.