A RESULT ON UNIQUENESS OF TRANSCENDENTAL MEROMORPHIC FUNCTIONS OF fnP(f) AND (fnP(f))(k) CONCERNING WEAKLY WEIGHTED SHARING

Abstract

In this article, we investigate the problem of transcendental meromorphic functions that share weak weights with one of their polynomials. Let $F$ and $G$ represents a transcendental meromorphic function. If $f^{n}P\left(f\right)-a_1$ and ${\left(f^{n}P\left(f\right)\right)}^{\left(k\right)}-a_2$ share “(0,1)” and $n,k$ and $m$ represent three positive integers such that $n\ge k+m+1$ then ${\left(f^{n}P\left(f\right)\right)}^{\left(k\right)}\equiv f^{n}P\left(f\right)a_2{a}_1^{-1}$. Furthermore, if $a_1\equiv a_2$, then $\mathcal{F}\left(z\right)=c{\mathrm{e}}^{z\lambda /n}$, where $c$ and $\lambda$ are nonzero constants such that ${\lambda }^{(k+m)}=1$. This result supports the Conjecture advanced by I. Lahiri, S. Majumder.