A note on the value distribution of $f^1(f^{(k)})^n$

Abstract

Let $f$ be a transcendental meromorphic function in the complex plane $\mathbf{C}$, and a be a nonzero complex number. We give quantitative estimates for the characteristic function $T(r,f)$ in terms of $N(r,1/( f^1(f^{(k)})^n-a))$, for integers $k$, $l$, $n$ greater than 1. We conclude that $f^1(f^{(k)})^n$ assumes every nonzero finite value infinitely often.