Abstract
In this study, the value distribution of the differential polynomial φ f 2 f ′ 2 − 1 is considered, where f is a transcendental meromorphic function, φ ( ≢ 0 ) is a small function of f by the reduced counting function. This result improves the existed theorems which obtained by Jiang (Bull Korean Math Soc 53: 365-371, 2016) and also give a quantitative inequality of φ f f ′ − 1 .
Highlights
Introduction and ResultsIn this paper, we assumed that the reader is familiar with the notations of Nevanlinna theory.Let f (z) and α(z) be two meromorphic functions in the complex plane
We assumed that the reader is familiar with the notations of Nevanlinna theory.Let f (z) and α(z) be two meromorphic functions in the complex plane
For any constant a in the complex plane we denote by Nk) (r, 1/( f − a)) the counting function of those a-points of f whose multiplicities are not great than k, by N(k (r, 1/( f − a)) the counting function of those a-points of f whose multiplicities are not less than k, by Nk (r, 1/( f − a)) the counting function of those a-points of f with multiplicity k, and denote the reduced counting function by N k) (r, 1/( f − a)), N (k (r, 1/( f − a)) and N k (r, 1/( f − a)), respectively
Summary
We assumed that the reader is familiar with the notations of Nevanlinna theory (see, e.g., [1,2]).Let f (z) and α(z) be two meromorphic functions in the complex plane. Reference [2] Let f (z) be a transcendental meromorphic function. D. Zhang proved the following two results: Theorem 2. (see [8]) Let f be a transcendental meromorphic function, α(6≡ 0, ∞) is a small function and δ(∞; f ) > 79 , f f 0 − α has infinitely many zeros. Bergweiler proved the following special case when f is of finite order and α is a polynomial: Theorem 4. (see [9]) If f is a transcendental meromorphic function of finite order and α is a non-vanishing polynomial, f f 0 − α has infinitely many zeros. Yu deals with the general situation of the small functions and proved the following result: Theorem 5. 1 can be reduced to 6 for n ≥ 2 in Theorem 7
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