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 following theorem: suppose that P P is a polynomial with deg ≥ 3 {\rm{\deg }}\hspace{0.33em}P\ge 3 f f transcendental meromorphic function. Let α \alpha be small function . If constant, also require there exists constant A ≠ A\ne such ( z ) − P\left(z)-A has zero multiplicity at least 3. Then, for any 0 < ε 1 0\lt \varepsilon \lt 1 , have xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> T r , ≤ k N ¯ + S T\left(r,f)\le k\overline{N}\left(r,\frac{1}{P(f)-\alpha }\right)+S\left(r,f), where if accent="false">′ P^{\prime} \left(z) only one zero, then = 2 k=\frac{1}{{\rm{\deg }}\hspace{0.33em}P-2} ; two distinct zeros a b b P\left(a)\ne P\left(b) nonconstant, k=\frac{1}{1-\varepsilon } otherwise k=1