Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts

Theorem 1.A

([7]) Let f(z) be a meromorphic function and let R(z)(≢ 0) be a rational function. Suppose that all but finitely many zeros and poles of f(z) are multiple. Then f′(z) – R(z) has infinitely many zeros. (In particular, if R(z) ≡ z, then f′(z) has infinitely many fixed points.)

The topic on fixed points can be also investigated in the field of complex differences. Here, the forward differences (see [8]) are defined by

where c ∈ ℂ∖{0}. For example, Chen and shon [9, 10, 11] have got some results on the zeros and fixed points of transcendental entire functions and meromorphic functions. Recently, Chen [12] and Zhang [13] studied the relationships between fixed points of meromorphic functions and their differences and shifts. Their results are stated as follows.

Theorem 1.B

([12]) Let f(z) be a finite order meromorphic function such that λ(1f(z))<ρ(f), and let c ∈ ℂ∖{0} be a constant such that f(z + c) ≢ f(z) + c. Then

Theorem 1.C

([13]) Let a ∈ ℂ and let f(z) be a finite order meromorphic function such that λ(f(z) – a) < ρ(f). Let c ∈ ℂ∖{0} be a constant. Then

Inspired by the previous results, especially Theorems 1.B and 1.C, we proceed to study the relationships between fixed points of meromorphic functions and their differences and shifts. Firstly, we consider higher order differences and shifts instead of first order ones, and obtain the following result.

Theorem 1.1

Let c ∈ ℂ∖{0}, n ∈ N₊, and let f(z) be a finite order transcendental meromorphic function. If f(z) has a Borel exceptional value a ∈ ℂ, then

Secondly, we generalize the case of fixed points into the more general case for n = 1, and obtain the following result.

Theorem 1.2

Let c ∈ ℂ∖{0}, m ∈ N₊, p(z) = p_mz^m + p_m–1z^m–1 + ⋯ + p₀ be a nonzero polynomial such that p_i ∈ ℂ, i = 0, 1, ⋯, m and p_m ≠ 0, and let f(z) be a finite order transcendental meromorphic function. If f(z) has a Borel exceptional value a ∈ ℂ, then

Lemma 2.1

([14]) Let f(z) be a meromorphic function with ρ(f) < ∞, and let η be a fixed nonzero complex number. Then for each ε > 0, we have

Lemma 2.2

([14]) Let f(z) be a meromorphic function with λ(1f(z))<∞, and let η be a fixed nonzero complex number. Then for each ε > 0, we have

Lemma 2.3

([13]) Let a ∈ ℂ and let f(z) be a meromorphic function with λ(f(z)–a) < ∞, and let η be a fixed nonzero complex number. Then for each ε > 0, we have

Lemma 2.4

([15]) Let A₀(z), ⋯, A_n(z) be entire functions of finite order such that among those having the maximal order ρ = max{ρ(A_k) : 0 ≤ k ≤ n}, exactly one has its type strictly greater than the others. Then for any meromorphic solution f(z) of

we have ρ(f) ≥ ρ + 1.

Lemma 2.5

([3]) Let f(z) be a meromorphic function. Then for all irreducible rational functions in f(z),

with meromorphic coefficients a_i(z), i = 0, 1, ⋯, p and b_j(z), j = 0, 1, ⋯, q such that

the characteristic function of R(z, f(z)) satisfies

Lemma 2.6

([13]) Suppose that h(z) is a nonconstant meromorphic function satisfying

Let

where a_i(z), i = 0, 1, ⋯, p, b_j(z), j = 0, 1, ⋯, q are small functions of h(z) and a₀b₀a_p ≢ 0. If q ≤ p and T(r, F) ≥ T(r, h) + S(r, h), then λ(F) = ρ(h).

Lemma 2.7

([2, 4]) Suppose that f₁(z), f₂(z), ⋯, f_n(z) are meromorphic functions and that g₁(z), g₂(z), ⋯, g_n(z) are entire functions satisfying the following conditions.

1. ∑j=1nfj(z)egj(z)≡0;

2. g_j(z) – g_k(z) are not constants for 1 ≤ j < k ≤ n;

3. for 1 ≤ j ≤ n, 1 ≤ h < k ≤ n,

   T(r,fj)=o{T(r,egh−gk)}n.e.asr→∞.

Then f_j(z) ≡ 0, j = 1, 2, ⋯, n.

Lemma 2.8

([13]) Let c be a nonzero constant, H(z) be a meromorphic function, and let h(z) be a polynomial with deg h(z) ≥ 1. If ρ(H) < ρ(e^h), then

Remark 2.9

From the proof of Lemma 2.8, we can also obtain

and

under the conditions in Lemma 2.8.

Proof of Theorem 1.1

As Theorem 1.C shows, the conclusions hold for n = 1. Next, we prove the conclusions for n ≥ 2.

Firstly, we prove the conclusions for n = 2. Suppose that τ(f(z)) < ρ(f), and we prove τ(Δc2f(z)) = τ(f(z+2c)) = ρ(f) next. Denote

Obviously, g(z) satisfies ρ(g₁) = ρ(f) < ∞. Then we have

and

which means that 0 and ∞ are Borel exceptional values of g₁(z). By Hadamard’s factorization theory, g₁(z) can be written as

where H₁(z)(≢ 0) is a meromorphic function such that ρ(H₁) < ρ(g₁) = ρ(f) and

is a polynomial such that ρ(f) = ρ(g₁) = deg h₁(z) = k, where k ∈ N₊ and a_i ∈ ℂ, i = 0, 1, ⋯, k, a_k ≠ 0. Substituting (3.2) into (3.1), we obtain

We get from (3.4) that

where

By (3.5), we obtain

where

By (3.5) and (3.6), we can see Δc2f(z) and Δc2f(z)−z as rational functions in e^h₁(z). Since ρ(H₁) < ρ(e^h₁) = k, by Lemma 2.8 and Remark 2.9, we get that the coefficients P_2,j(z), j = 1, 2, ⋯, 7 are small functions with respect to e^h₁(z), that is, T(r, P_2,j(z)) = S(r, e^h₁(z)), j = 1, 2, ⋯, 7. Next, we assert P_2,1(z) ≢ 0. Since

we get from k ∈ N₊, a_k ≠ 0 and c ≠ 0 that

and

Thus, if P_2,1(z) ≡ 0, then by Lemma 2.4, we have ρ(H₁) ≥ k – 1 + 1 = k, which contradicts with ρ(H₁) < ρ(g₁) = ρ(f) = k. So, the assertion P_2,1(z) ≢ 0 holds. Consequently, by Lemma 2.7 and P_2,1(z) ≢ 0, we have

Then by (3.5) and the obvious fact that P_2,5(z) ≢ 0, we have

and consequently

By (3.6), (3.7) and Lemma 2.6, we have

that is,

Meanwhile, we have by (3.4) that

where (z – a)H₁(z + 2c) + 2cH₁(z + 2c) = (z – a + 2c)H₁(z + 2c) ≢ 0. Thus, f(z + 2c) – z can be seen as an irreducible rational function in e^h₁(z+2c). Since ρ(H₁(z + 2c)) = ρ(H₁) < ρ(e^h₁) = ρ(e^h₁(z+2c)), by Lemma 2.8 and Remark 2.9, we have

By (3.9), (3.10) and Lemma 2.5, we have

Then, by Lemma 2.6, we have

that is,

Suppose that τ(f(z + 2c)) < ρ(f), and we prove τ(Δc2f(z))=ρ(f) next. Denote

From (3.11) we get

and

By (3.11), (3.12) and Lemma 2.1, we have ρ(g₂) = ρ(g₂(z – 2c)) = ρ(f(z + 2c)) = ρ(f). By Lemma 2.3 and the assumption that λ(f(z)–a) < ρ(f), we have

and

which means that 0 and ∞ are Borel exceptional values of g₂(z). Then following the steps similar to (3.2)-(3.8), we have τ(Δc2f(z))=λ(Δc2f(z)−z)=ρ(f), that is,

Secondly, we prove the conclusions for n ≥ 3. Suppose that τ(f(z)) < ρ(f), and we prove τ(Δcnf(z)) = τ(f(z + nc)) = ρ(f) next. By (3.1), (3.2) and (3.4), we have

where

By (3.13), we have

By (3.13) and (3.14), We can see Δcnf(z) and Δcnf(z)−z as rational functions in e^h₁(z). By Lemma 2.8 and Remark 2.9, we have

So, the coefficients P_n,j(z), j = 1, 2, ⋯, 2n + 1 are small functions with respect to e^h₁(z). Next, we assert P_n,1(z) ≢ 0. We rewrite P_n,1(z) as

Clearly, we have

and

Obviously, ∑i=0n−1(−1)icni(z+(n−i)c−a)=∑i=0n(−1)icni(z+(n−i)c−a)−(−1)n(z−a)=(−1)n+1(z−a)≢0. Thus, if P_n,1(z) ≡ 0, then by Lemma 2.4, we have ρ(H₁) ≥ k – 1 + 1 = k, which contradicts with ρ(H₁) < k. So, the assertion P_n,1(z) ≢ 0 holds. Consequently, by Lemma 2.7 and P_n,1(z) ≢ 0, we have

Then by (3.13) and the obvious fact that P_n,2n+1(z) ≢ 0, we have

and consequently