Value Distribution of Differential Difference Polynomials

Abstract

Though the Value Distribution Theory of Nevanlinna is about a century old, it is an active area of research. It has a wide range of applications within and outside function theory. In the nineteenth century, the famous mathematician E. Picard obtained the path-breaking result on value distribution. Later, E. Borel, introduced the concept of order of an entire function. Picard Borel Theorem, laid the foundation for the theory of value distribution and since then it has been the source of many research accomplishments on this subject. However, we aim to concentrate on some areas of value distribution theory. The concepts of distribution of zeros to the Hayman conjecture are extended. This work also focuses on the study of uniqueness results, and differential-difference equations of entire( meromorphic) functions. In addition to this, analysing the deficiencies of entire and meromorphic functions and extending and generalizing the results on set sharing are also considered. Placed in this context, the research explores the existing literature, techniques and methodology which, in a way, gives an overview. The basis of the value distribution theory of meromorphic functions consists of several formulae connecting the behaviour of a meromorphic function with the distribution of its Iv zeros and poles. In the study of value distribution of meromorphic functions, one of the important topics is to characterize meromorphic functions in terms of points at which they assume some values. The various characteristics of meromorphic functions are the main tool in the study of value distribution of meromorphic functions. Over the last few decades mathematicians are putting their effort into studying differential equations, differential-difference equations and difference equations in the complex plane by relying on the Nevanlinna theory. Recent work is focused on the existence and the growth of meromorphic solutions of Fermat-type equations. Many results are obtained by extending and improving the previous results.