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

TL;DR

This paper provides quantitative estimates for the value distribution of a specific class of transcendental meromorphic functions, showing they assume every nonzero finite value infinitely often.

Contribution

It introduces new bounds relating the characteristic function to counting functions for functions of the form $f^l(f^{(k)})^n$, advancing understanding of their value distribution.

Findings

Quantitative estimates for $T(r,f)$ in terms of $N(r,1/(f^l(f^{(k)})^n - a))$

Proof that $f^l(f^{(k)})^n$ takes every nonzero finite value infinitely often

Extension of value distribution theory to complex functions of this form

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.