In 2010, Li-Ye [13, Theorem 0.1] proved that P(ζ(z), ζ'(z), . . . , ζ(m)(z), Γ(z), Γ'(z), Γ"(z)) ≢ 0 in ℂ, where m is a non-negative integer, and P(u0, u1, . . . , um, v0, v1, v2) is any non-trivial polynomial in its arguments with coefficients in the field ℂ. Later on, Li-Ye [15, Theorem 1] proved that P(z, Γ(z), Γ'(z), . . . , Γ(n)(z), ζ(z)) ≢ 0 in z ∈ ℂ for any non-trivial distinguished polynomial P(z, u0, u1, . . ., un, v) with coefficients in a set Lδ of the zero function and a class of nonzero functions f from ℂ to ℂ ∪ {∞} (cf. [15, Definition 1]). In this paper, we prove that P(z, ζ(z), ζ'(z), . . . , ζ(m)(z), Γ(z), Γ'(z), . . . , Γ(n)(z)) ≢ 0 in z ∈ ℂ, where m and n are two non-negative integers, and P(z, u0, u1, . . . , um, v0, v1, . . . , vn) is any non-trivial polynomial in the m + n + 2 variables u0, u1, . . . , um, v0, v1, . . . , vn with coefficients being meromorphic functions of order less than one, and the polynomial P(z, u0, u1, . . . , um, v0, v1, . . . , vn) is a distinguished polynomial in the n + 1 variables v0, v1, . . . , vn. The question studied in this paper is concerning the conjecture of Markus from [16]. The main results obtained in this paper also extend the corresponding results from Li-Ye [12] and improve the corresponding results from Chen-Wang [5] and Wang-Li-Liu-Li [23], respectively.