Uniqueness of differential polynomial of meromorphic function with its $q$-shift - Gurudas, Biswas (Matematychni Studii, 2025)
In the paper, we apply the concept of weighted sharing to study the uniqueness problems of differential polynomial of meromorphic function of zero order with its $q$-shift. The results of the paper improve and extend some recent results due to H. P. Waghamore and M. M. Manakame [Int. J. Open Problems Compt. Math., 18 (2025), 22-34]. A typical theorem obtained in the paper is as follows: Let $P$ be a polynomial, $f(z)$ be a non-constant meromorphic function of zero-order. Suppose that $q$ is a non-zero complex constant, $η \in \mathbb{C}$ and $n$ is an integer satisfying $n≥ m+3τ+3Ω +6$, where $m=\deg P$, $τ =\sum_{j=1}^{s}μ _{j}$ and $Ω =\sum_{j=1}^{s}jμ _{j}.$If $f^{n}(z)P(f(z))\prod _{j=1}^{s}f^{(j)}(z)^{μ _{j}}$ and $f^{n}(qz+η )P(f(qz+η ))\prod _{j=1}^{s}f^{(j)}(qz+η )^{μ _{j}}$ share $(1,2)$ and $(∞,∞)$, then $\displaystyle f^{n}(z)P(f(z))\prod _{j=1}^{s}f^{(j)}(z)^{μ _{j}}\equiv f^{n}(qz+η )P(f(qz+η ))\prod _{j=1}^{s}f^{(j)}(qz+η )^{μ _{j}}.$ Three other similar theorems are also obtained in the paper.