Paired Hayman conjecture of some delay-differential polynomials that share a small function

In this paper, we investigate the uniqueness problems of P(f)L(g) and P(g)L(f) when they share a nonzero small function α(z) with finite weights. Here L(h) may take the derivatives h(k)(z) or the shift h(z+c) or the difference h(z+c)-h(z) or the delay-differential h(k)(z+c), k>=1, and c is a nonzero constant and P(z) is a polynomial of degree n. Also, f(z) and g(z) are transcendental meromorphic (or entire) functions and α(z) is a small function with respect to both f(z) and g(z). The results of the paper improve and supplement the recent results of Sahoo and Pal [17].