Some results on the value distribution of differential polynomials

In this article, we study some results on the value distribution of differential polynomials and mainly prove the following theorem: suppose that P is a polynomial with deg P >= 3 and f is a transcen- dental meromorphic function. Let alpha be a small function of f. If alpha is a constant, we also require that there exists a constant A &NOTEQUexpressionL; alpha such that P(z) -A has a zero of multiplicity at least 3. Then, for any 0 < epsilon < 1, we have Tr,f <= kN (1/r, P f-alpha )+ S(r,f),where if P '(z) has only one zero, then k = 1/deg p- 2 ; if P '(z) has two distinct zeros a and b with P(a) &NOTEQUexpressionL;P( b) 1 k P- and alpha is nonconstant, then k = 1/1-epsilon otherwise k = 1