Hayman conjecture and uniqueness of some delay-differential polynomials sharing a small function

In this paper, we investigate the uniqueness problem of P(f)L(g) and P(g)L(f) when they share $$\alpha (z)$$ counting multiplicities, where L(h) represents any one of $$h^{(k)}(z),\; h(z+c),\; h(z+c)-h(z)$$ and $$h^{(k)}(z+c)$$ where k is a positive integer, c is a nonzero constant and $$\alpha (z)$$ is a nonzero small function with respect to both f(z) and g(z). The results of the paper generalize the recent results of Gao and Liu (Bull Korean Math Soc 59:155–166, 2022).