Abstract

In this paper, we study the uniqueness problems of $f^n(z)L(g)$ and $g^n(z)L(f)$ when they share a non-zero small function $\alpha(z)$ with finite weights, where $L(h)$ represents any one of $h^{(k)}(z),\; h(z+c), \;h(z+c)-h(z)$ and $h^{(k)}(z+c),$ $k\geq 1$ and $c$ is a non-zero constant. Here $f(z)$ and $g(z)$ are transcendental meromorphic (or entire) functions and $\alpha(z)$ is a small function with respect to both $f(z)$ and $g(z).$ Our results improve and supplement the recent results due to Gao and Liu [Bull. Korean Math. Soc. 59 (2022), 155-166].

Keywords: Uniqueness; Hayman conjecture; delay-differential polynomial; difference polynomial; weighted sharing.

MSC: 30D35, 39A05

DOI: 10.57016/MV-L10OEQ79

Pages:  1$-$15