On the value distribution of a Differential Monomial and some normality criteria

2019/03/24 by Weiran Lü, Lü, Weiran, Bikash Chakraborty +1
Mathematics · #Meromorphic and Entire Functions #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1903.10940

Abstract

Let f be a transcendental meromorphic function defined in the complex plane ℂ, and φ(\not≡ 0,∞) be a small function of f. In this paper, We give a quantitative estimation of the characteristic function T(r, f) in terms of N(r,(1)/(M[f]-φ(z))) as well as \olN(r,(1)/(M[f]-φ(z))), where M[f] is the differential monomial, generated by f.\par Moreover, we prove one normality criterion: Let \mathscrF be a family of analytic functions on a domain D and let k(≥1), q₀(≥ 3), q_i(≥0) (i=1,2,…,k-1), q_k(≥1) be positive integers. If for each f∈ \mathscrF, f has only zeros of multiplicity at least k, and f^q₀(f')^q₁...(f^(k))^q_k\not=1, then \mathscrF is normal on domain D.

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