With the and of the notion of weighted sharing we prove some uniqueness theorems of meromorphic functions which improve some earlier results

Weighted value sharing and uniqueness of meromorphic functions

Preface Introduction 1. Quasiconformal geometry 2. Extensions and interpolations 3. Preliminaries on dynamical systems and actions of Kleinian groups 4. Introduction to surgery and first occurrences 5. General principles of surgery 6. Soft surgeries with a contribution by X. Buff and C. Henriksen 7. Cut and paste surgeries with contributions by K. M. Pilgrim, Tan Lei and S. Bullett 8. Cut and paste surgeries with sectors with a contribution by A. L. Epstein and M. Yampolsky 9. Trans-quasiconformal surgery with contributions by C. L. Petersen and P. Haissinsky Bibliography Symbol index Index.

Quasiconformal surgery in holomorphic dynamics

In this paper, we study the uniqueness of entire functions and prove the following theorem. Let f ( z ) and g ( z ) be two nonconstant entire functions, n , k two positive integers with n > 2 k + 4. If [ f n ( z )] ( k ) and [ g n ( z )] ( k ) share 1 with counting the multiplicity, then either f ( z ) = c 1 e cz , g ( z ) = c 2 e − cz , where c 1 , c 2 , and c are three constants satisfying (−1) k ( c 1 c 2 ) n ( nc ) 2 k = 1, or f ( z ) ≡ tg ( z ) for a constant t such that t n = 1.

Uniqueness and value-sharing of entire functions☆

We discuss the uniqueness problem of meromorphic functions sharing one value and obtain two theorems which improve a result of Xu and Qu and supplement some other results earlier given by Yang, Hua, and Lahiri.

https://downloads.hindawi.com/journals/ijmms/2005/746476.pdf

Meromorphic functions sharing one value

Value distribution and shared sets of differences of meromorphic functions

Concerning the uniqueness and sharing values of meromorphic functions, many results about meromorphic functions that share more than or equal to two values have been ob- tained. In this paper, we shall study meromorphic functions that share only one value, and prove the following result: For n ≥ 11 and two meromorphic functions f(z) and g(z), if f n fand g n g ' share the same nonzero and finite value a with the same multiplicities, then f ≡ dg or g = c1e cz and f = c2e −cz , where d is an (n + 1)th root of unity, c, c1 and c2 being constants. As an application, we solve some non-linear differential equations.

/pdf/uniqueness-and-value-sharing-of-meromorphic-functions-4rqu60p569.pdf

Uniqueness and value-sharing of meromorphic functions

Let f be a nonconstant meromorphic function. The sequence of the iterates of f is denoted by 

$${f^0}=id,{f^1}=f,\cdots,{f^{n + 1}} = {f^n}(f), \cdots$$

Dynamics of Transcendental Functions

https://projecteuclid.org/download/pdf_1/euclid.rmjm/1181069360

Dynamics of Permutable Transcendental Entire Functions

This paper is concerned with the dynamics of transcendental entire functions. Let f.z/ be a transcendental entire function. We shall study the boundedness of the components of the Fatou setF.f/under some restrictions on the growth of the function. This relates to a problem due to Baker in 1981.

/pdf/fatou-components-of-entire-functions-of-small-growth-1i2d9k85xm.pdf

Fatou components of entire functions of small growth

For any transcendental meromorphic function, we shall study the relations between the components of the Fatou set and their images. This relates to a problem studied by Bergweiler [1993].

Xinhou Hua

Papers

Uniqueness and value-sharing of meromorphic functions

Dynamics of Transcendental Functions

Dynamics of Permutable Transcendental Entire Functions

Fatou components of entire functions of small growth

Fatou Components and a Problem of Bergweiler