Characteristic estimation of differential polynomials

Abstract

Abstract In this paper, we give the characteristic estimation of a meromorphic function f with the differential polynomials f l ( f ( k ) ) n $f^{l}(f^{(k)})^{n}$ and obtain that T ( r , f ) ≤ M N ‾ ( r , 1 f l ( f ( k ) ) n − a ) + S ( r , f ) $$\begin{aligned} T(r,f)\leq M\overline{N} \biggl(r,\frac{1}{f^{l}(f^{(k)})^{n}-a} \biggr)+S(r,f) \end{aligned}$$ holds for M = min { 1 l − 2 , 6 } $M=\min \{\frac{1}{l-2},6\}$ , integers l ( ≥ 2 ) $l(\geq 2)$ , n ( ≥ 1 ) $n(\geq 1)$ , k ( ≥ 1 ) $k(\geq 1)$ , and a non-zero constant a. This quantitative estimate is an interesting and complete extension of earlier results. The value distribution of a differential monomial of meromorphic functions is also investigated.

Keywords

• Nevanlinna theory
• Value distribution
• Meromorphic solution
• Nevanlinna characteristic