Uniqueness of $p(f)$ and $P[f]$

Authors: KULDEEP SINGH CHARAK, BANARSI LAL

Abstract: Let $f$ be a nonconstant meromorphic function, $a (\not\equiv 0, \infty)$ be a meromorphic function satisfying $T(r,a) = o(T(r,f))$ as $r \rightarrow \infty$, and $p(z)$ be a polynomial of degree $n \geq 1$ with $p(0) = 0$. Let $P[f]$ be a nonconstant differential polynomial of $f$. Under certain essential conditions, we prove that $p(f) \equiv P[f]$, when $p(f)$ and $P[f]$ share $a$ with weight $l \geq 0$. Our result generalizes the results due to Zhang and L$\ddot{\text{u}}$, Banerjee and Majumdar, and Bhoosnurmath and Kabbur and answers a question asked by Zhang and L$\ddot{\text{u}}$.

Keywords: Meromorphic functions, small functions, sharing of values, differential polynomials, Nevanlinna theory

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