Let (Q_{d} z,f)) be a differential polynomial in f of degree d with rational function coefficients.
Let ℱ be a family of meromorphic functions defined in D, k and q (≥2) be two positive integers, and H ( f, f ′, …, f ( k ) ) be a differential polynomial with Γ γ | H < k + 1.
Let k and q (≥2) be two positive integers, b ≠ 0 be a complex number, and let H ( f, f ′, …, f ( k ) ) be a differential polynomial with Γ γ | H < k + 1.
Theorem 1.4 Let k be a positive integer, b (≠0) be a complex number, h ( z ) be a polynomial, and let H ( f, f ′, …, f ( k ) ) be a differential polynomial with Γ γ | H < k + 1.
Theorem 1.2 Let ℱ be a family of meromorphic functions defined in D, k be a positive integer, let h ( z ) be a polynomial with degree at least 2, and H ( f, f ′, …, f ( k ) ) be a differential polynomial with Γ γ | H < k + 1.
Let w z) be an entire function in the complex plane and all zeros of w z) have multiplicity at least k (k ∈ ℕ), P[w] be a differential polynomial with constant coefficients in variable w or degz,∞a t ≤ 0(t ∈ I) in the (1.2) and nkq > deg P[w] (n ∈ ℕ).
Substituting these in (28) shows that A is a differential polynomial in C'.
where m, n ∈ N, a ≠ 0, m is an odd integer, and P [ w ] is a differential polynomial.
where P ̃ 2 k - 1 is a differential polynomial in α' of degree at most 2k - 1. From (3.14).
Then, P = - C ′ 2 + 2 C ″ + 2 C ′ 2 + k 2 e - 2 C = 2 C ″ + C ′ 2 + k 2 e - 2 C. Substituting these in (28) shows that A is a differential polynomial in e-Cand C'.
If (R z)) replaced by (R z,f)) in the theorem, where (R z,f)) is a differential polynomial in f with rational function as its coefficients, then the conclusion is not true generally.
