Abstract
For a transcendental meromorphic function f ( z ) , the main aim of this paper is to investigate the properties on the zeros and deficiencies of some differential-difference polynomials. Some results about the deficiencies of some differential-difference polynomials concerning Nevanlinna deficiency and Valiron deficiency are obtained, which are a generalization of and improvement on previous theorems given by Liu, Lan and Zheng, etc.
Highlights
Nevanlinna established the famous Nevanlinna theory, which is an important tool in studying the value distribution of meromorphic functions in Complex Analysis
After several decades or even hundreds of years of development, a lot of interesting and important results exist on the value distribution of meromorphic functions
This article is devoted to the study of value distribution of some differential-difference polynomials of meromorphic function concerning the Nevanlinna and Valiron exceptional values
Summary
In 1970s, Yang [13,14] further investigated this problem and extended the results to some differential polynomial in f (z), when f (z) is a transcendental meromorphic function satisfying. Let f (z) be a transcendental meromorphic function of finite order, and let P(z, f )(6≡ 0) be a differential-difference polynomial of the form (2), with n different shifts. Let f (z) be a transcendental meromorphic function of finite order satisfying N (r, f ) = S(r, f ), and let P(z, f )(6≡ 0) be a differential-difference polynomial of the form (2), with n different shifts. By applying Theorems 6–9 to Q3 (z, f ), we conclude that Q2 (z, f ) has infinitely many zeros and δ(0, Q2 (z, f )) < 1
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