Some results on the value distribution of differential polynomials

Abstract

In this article, we study some results on the value distribution of differential polynomials and mainly prove the following theorem: suppose that PP is a polynomial with degP≥3{\rm{\deg }}\hspace{0.33em}P\ge 3 and ff is a transcendental meromorphic function. Let α\alpha be a small function of ff. If α\alpha is a constant, we also require that there exists a constant A≠αA\ne \alpha such that P(z)−AP\left(z)-A has a zero of multiplicity at least 3. Then, for any 0<ε<10\lt \varepsilon \lt 1, we have T(r,f)≤kN¯r,1P(f)−α+S(r,f),T\left(r,f)\le k\overline{N}\left(r,\frac{1}{P(f)-\alpha }\right)+S\left(r,f), where if P′(z)P^{\prime} \left(z) has only one zero, then k=1degP−2k=\frac{1}{{\rm{\deg }}\hspace{0.33em}P-2}; if P′(z)P^{\prime} \left(z) has two distinct zeros aa and bb with P(a)≠P(b)P\left(a)\ne P\left(b) and α\alpha is nonconstant, then k=11−εk=\frac{1}{1-\varepsilon }; otherwise k=1k=1.

Keywords

• meromorphic function
• small function
• deficiency value
• 30d35