UNIQUENESS OF CERTAIN DIFFERENTIAL POLYNOMIALS WITH FINITE WEIGHT

Document Type : Regular research papers

Authors

1 Presidency Unviersity, Bengaluru Presidency Unviersity

2 Assistant Professor Presidency University Bengaluru

3 Presidency Unviersity, Bengaluru

Abstract

Some fundamental terms in Nevanlinna’s value distribution theory $m(r, f)$, $N(r, f)$,
$T(r, f)$, etc. and let $f(z)$ and $g(z)$ be two non-constant meomorphic functions, $P(f)$ and $P(g)$ be a polynomials of degree $m$, whose zeros and poles are of multiplicities atleast $s$, where $s$ is a positive integer. and let $n$, $k$ be two positive integers with $s(n+m)>9k+14$. If $m\geq{2}$ and $\delta(\infty,f)>\dfrac{2+d}{n+m}$, if $m=1$ and $\Theta(\infty,f)>\dfrac{2+d}{n+1}$, $[f^{n}P(f)]^{(k)}$ and $[g^{n}P(g)]^{(k)}$ share $1(1,0)$, then either $[f^{n}P(f)]^{(k)} [g^{n}P(g)]^{(k)}\equiv{1}$ or $f(z)$ and $g(z)$ satisfy the algebraic equation $R(f,g)=0$, where
\begin{equation*}
R(\omega_{1},\omega_{2})=\omega^{m}_{1}(a_{m}\omega^{m}_{1}+a_{m-1}\omega^{m-1}_{1}+...+a_{0})-\omega^{m}_{2}(a_{m}\omega^{m}_{2}+a_{m-1}\omega^{m-1}_{2}+...+a_{0}).
\end{equation*}
Let $f(z)$ and $g(z)$ be two non-constant entire functions with satisfying inequality $n>5k+6m+7$. The present paper deals with the study of uniqueness of certain differential polynomials with the notion of weighted sharing. The results of the paper improve and generalize the results of Rajeshwari S, Husna V and Nagarjun V.\cite{06}. We have also exhibited a series of examples satisfying our results and provided some other examples showing the sharpness of one of our results.

Keywords

Files

Share

How to cite

Statistics