On the value distribution of the differential polynomial $Af^{n}f^{(k)}+Bf^{n+1}-1$

Abstract

In the paper, we study the value distribution of the differential polynomial $Af^{n}f^{(k)}+Bf^{n+1}-1$, where $f(z)$ is a transcendental meromorphic function and $n (\geq 2), k(\not= 2)$ are positive integers. We prove an inequality for the Nevanlinna characteristic function $T(r,f)$  in terms of  reduced counting function only. The result of the paper not only improves the result due to Q.D. Zhang [J. Chengdu Ins. Meteor., 20(1992), 12-20], also partially improves a recent result of H. Karmakar and P. Sahoo [Results Math., (2018),73:98].