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

Abstract

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