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Normal families of meromorphic functions sharing values or functions
Journal of Inequalities and Applications volume 2011, Article number: 72 (2011)
Abstract
In this paper, we investigate the normal families of meromorphic functions concerning shared values and shared analytic functions and prove some normal criteria that generalize or extend some results obtained by Q. C. Zhang, Y. T. Li and Y. X. Gu, J. M. Chang.
Mathematics Subject Classification (2000): 30D45; 30D35.
1. Introduction and main results
The notations and concepts used in this paper can be found in [1–3]. In this paper, We also use f(z) = a ⇒ g(z) = b to stand for g(z) = b when f(z) = a.
Let D be a domain in the complex plane C, \mathcal{F} be a family of meromorphic functions defined in D. \mathcal{F} is said to be normal in D, in the sense of Montel, if every sequence {f}_{n}\left(z\right)\in \mathcal{F}\left(n\phantom{\rule{2.77695pt}{0ex}}=1,2,\phantom{\rule{2.77695pt}{0ex}}\dots \right) has a subsequence {f}_{{n}_{k}}\left(z\right)\left(k=1,2,\phantom{\rule{2.77695pt}{0ex}}\dots \right) that converges spherically locally uniformly in D, to a meromorphic function or ∞(see [2, 4, 5]).
In 1998, Y. F. Wang and M. L. Fang [6] proved the following theorem.
Theorem A. Let \mathcal{F} be a family of meromorphic functions in D, n, k ∈ N with n ≥ k + 2. If for every function f\in \mathcal{F}, f has only zeros of order at least n and f^{(k)}≠ 1, then \mathcal{F} is normal in D.
In 2004, M. L. Fang and L. Zalcman [7] proved the following theorem.
Theorem B. Let \mathcal{F} be a family of meromorphic functions in D, and n be a positive integer. If for each pair of functions f and g in \mathcal{F}, f and g share the value 0 and f^{n}f' and g^{n}g' share a nonzero value b in D, then \mathcal{F} is normal in D.
In 2008, Q. C. Zhang [8] proved the following Theorems C and D, which generalized the condition f^{(k)}≠ 1 in Theorem A to shared value when k = 1 and generalized Theorem B when n ≥ 2, respectively.
Theorem C. Let \mathcal{F} be a family of meromorphic functions in D satisfying that all of zeros and poles of f\in \mathcal{F} have multiplicities at least 3. If for each pair of functions f and g in \mathcal{F}, f' and g' share a nonzero value b in D, then \mathcal{F} is normal in D.
Theorem D. Let \mathcal{F} be a family of meromorphic functions in D, and n be a positive integer. If n ≥ 2 and for each pair of functions f and g in \mathcal{F}, f^{n} f' and g^{n}g' share a nonzero value b in D, then \mathcal{F} is normal in D.
In 2009, Y. T. Li and Y. X. Gu [9] proved the following theorem with high order derivatives.
Theorem E. Let \mathcal{F} be a family of meromorphic functions defined in a domain D. Let k, n ≥ k + 2 be positive integers and a ≠ 0 be a finite complex number. If (f^{n} )^{(k)}and (g^{n} )^{(k)}share a in D for every pair of functions f,\phantom{\rule{2.77695pt}{0ex}}g\in \mathcal{F}, then \mathcal{F} is normal in D.
In this paper, we investigate the normal family of meromorphic functions with higher order derivatives and obtain the following two theorems, which generalize Theorems C, D and E.
Theorem 1.1. Suppose that d(≥ 0) is an integer, p(z) is an analytic function in D, and the multiplicity of its all zeros is at most d. Let \mathcal{F} be a family of holomorphic functions in D; the multiplicity of all zeros of f\in \mathcal{F} is at least k + d + 2. If for each pair of functions f and g in \mathcal{F}, f^{(k)}and g^{(k)}share p(z) in D, then \mathcal{F} is normal in D.
Theorem 1.2. Suppose that d(≥ 0) is an integer, p(z) is an analytic function in D, and the multiplicity of its all zeros is at most d. Let \mathcal{F} be a family of meromorphic functions in D, the multiplicity of all zeros and poles of f\in \mathcal{F} is at least k + 2d + 2. If for each pair of functions f and g in \mathcal{F}, f^{(k)}and g^{(k)}share p(z) in D, then \mathcal{F} is normal in D.
Corollary 1.1. Let \mathcal{F} be a family of meromorphic functions in D; the multiplicity of all zeros and poles of f\in \mathcal{F} is at least k + 2. a ≠ 0 is a finite complex number. If for each pair of functions f and g in \mathcal{F}, f^{(k)}and g^{(k)}share a in D, then \mathcal{F} is normal in D.
Corollary 1.2. Suppose that d(≥ 0) is an integer, p(z) is an analytic function in D, and the multiplicity of its all zeros is at most d. Let \mathcal{F} be a family of holomorphic functions in D, n be a positive integer. If n ≥ d + 2 and for each pair of functions f and g in \mathcal{F}, f^{n}f' and g^{n}g' share p(z) in D, then \mathcal{F} is normal in D.
Corollary 1.3. Suppose that d(≥ 0) is an integer, p(z) is an analytic function in D, and the multiplicity of its all zeros is at most d. Let \mathcal{F} be a family of meromorphic functions in D, n be a positive integer. If n ≥ 2d + 2 and for each pair of functions f and g in \mathcal{F}, f^{n}f' and g^{n} g' share p(z) in D, then \mathcal{F} is normal in D.
Remark 1.1. From Corollary 1.1, we can deduce Theorems C and E; from Corollary 1.3, we can deduce Theorem D.
About the normality concerning shared value of meromorphic functions with its derivatives, J. M. Chang [10] recently obtained the following theorem.
Theorem F. Let \mathcal{F} be a family of meromorphic functions in D, a, b be two distinct finite nonzero complex numbers. If for every f\in \mathcal{F}, f(z) = a ⇒ f'(z) = a, f'(z) ≠ b, and f"(z) ≠ b, then \mathcal{F} is normal in D.
Thus, a natural question is: whether \mathcal{F} is normal if we replace the condition f(z) = a ⇒ f'(z) = a in Theorem F by f'(z) = a ⇒ f(z) = a. We answer this question by the following result.
Theorem 1.3. Let \mathcal{F} be a family of meromorphic functions in D, a, b be two distinct finite complex numbers. If for every f\in \mathcal{F}, f'(z) = a ⇒ f(z) = a, f'(z) ≠ b, and all of its zeros have multiplicity at least 2, then \mathcal{F} is normal in D.
2. Lemmas
Lemma 2.1. Let d(≥ 0), k(≥ 1) be two integers, p\left(z\right)={a}_{d}{z}^{d}+{a}_{d1}{z}^{d1}+\cdots +{a}_{1}z+{a}_{0} be a polynomial, where a_{ d } (≠ 0), a_{d1}, ..., a_{0} are constants. If g(z) is a nonconstant polynomial, the multiplicity of its all zeros is at least k + d + 2, then g^{(k)}(z)  p(z) has at least two distinct zeros, and g^{(k)}(z)  p(z) ≢ 0.
Proof We discuss in two cases:
Case 1. If g^{(k)}(z)  p(z) ≠ 0, then g^{(k)}(z)  p(z) ≡ C, where C is a constant. So g\left(z\right)=\frac{{a}_{d}{z}^{d+k}}{\left(d+k\right)\left(d+k1\right)\cdots \left(d+1\right)}+\frac{{a}_{d1}{z}^{d+k1}}{\left(d+k1\right)\left(d+k2\right)\cdots d}+\cdots +\frac{{a}_{0}{z}^{k}}{k\left(k1\right)\cdots 1}+q\left(z\right), where q^{(k)}(z) ≡ C, i.e., q(z) is a polynomial of degree at most k, then g(z) is a polynomial of degree k + d, which contradicts with that the multiplicity of all zeros of g(z) is at least k + d + 2.
Case 2. If g^{(k)}(z)  p(z) has only one zero ξ_{0}, we assume g^{(k)}(z)  p(z) = A(z  ξ_{0}) ^{r} , where A is a nonzero constant, r is a positive integer.
We discuss the following three cases:

(i)
If r < d + 1, then g(z) is a nonconstant polynomial of degree at most k + d, which contradicts with that the multiplicity of all zeros of g(z) is at least k + d + 2.

(ii)
If r = d + 1, then g ^{(k+d+1)}(z) = A·r·(r  1) ····· 2·1, so g ^{(k+d+1)}(z) has no zero. Since the multiplicity of all zeros of g(z) is at least k + d + 2, then g(z) has no zero, which contradicts with g(z) is a nonconstant polynomial.

(iii)
If r > d + 1, then g ^{(k+d)}(z)  a_{ d } · d · (d  1) ····· 2·1 = A · r · (r  1) ····· (r  d + 1) (z  ξ _{0})^{rd}, g ^{(k+d+1)}(z) = A · r · (r  1) ····· (r  d) (z  ξ _{0})^{rd1}, so ξ _{0} is the unique zero of g ^{(k+d+1)}(z). Since g(z) is a nonconstant polynomial and the multiplicity of all zeros of g(z) is at least k + d + 2, then ξ _{0} is a zero of g, thus, g ^{(k+d)}(ξ _{0}) = 0, which contradicts with g ^{(k+d)}(ξ _{0}) = a_{ d } · d · (d  1) ····· 2·1 ≠ 0.
From Case 1 and Case 2, we know g^{(k)}(z)  p(z) has at least two distinct zeros.
If g^{(k)}(z)  p(z) ≡ 0, then similar to the proof of Case 1, we obtain that g(z) is a polynomial of degree k + d and get a contradiction since that the multiplicity of all zeros of g(z) is at least k + d + 2. Then, g^{(k)}(z)  p(z) ≢ 0.
Lemma 2.2. Let d(≥ 0), k(≥ 1) be two integers, p(z) = a_{ d } z^{d} + a_{d1}z^{d1}+ · · · + a_{1}z + a_{0} be a polynomial, where a_{ d } (≠ 0), a_{d1}, ..., a_{0} are constants. If g(z) is a rational function and not a polynomial, and the multiplicity of all the zeros and poles of g(z) is at least k + 2d + 2, then g^{(k)}(z)  p(z) has at least two distinct zeros, and g^{(k)}(z)  p(z) ≢ 0.
Proof Since g(z) is a rational function and not a polynomial, then obviously g^{(k)}(z)  p(z) ≢ 0. Let
where B is a nonzero constant, γ_{ i } (i = 1, 2, ..., n) and δ_{ j } (j = 1, 2, ..., m) are the zeros and poles of g(z), their multiplicity are p_{ i } (i = 1, 2, ..., n) and q_{ j } (j = 1, 2, ..., m), respectively. Then, p_{ i } , q_{ j } ≥ k + 2d + 2(i = 1, 2, ..., n, j = 1, 2, ..., m).
Differentiating both sides of (2.1) step by step, we have
where {h}_{k}\left(z\right)=\left(pq\right)\left(pq1\right)\cdots \left(pqk+1\right){z}^{k\left(n+m1\right)}+\cdots +{c}_{k}^{\left(1\right)}z+{c}_{k}^{\left(0\right)} is a polynomial, {c}_{k}^{\left(i\right)}\left(i=0,1\right) are constants.
where {h}_{k+d+1}\left(z\right)=\left(pq\right)\left(pq1\right)\cdots \left(pqkd\right){z}^{\left(k+d+1\right)\left(n+m1\right)}+\cdots +{c}_{k+d+1}^{\left(1\right)}z+{c}_{k+d+1}^{\left(0\right)} is a polynomial, {c}_{k+d+1}^{\left(i\right)}\left(i=0,\phantom{\rule{2.77695pt}{0ex}}1\right) are constants.
Next, we discuss in two cases.
Case 1. If g^{(k)}(z)  p(z) has a unique zero ξ_{0}, then let
where D is a nonzero constant and l is a positive integer.
Here, we discuss in two subcases.
Subcase 1.1. When d ≥ l.
Differentiating both sides of (2.4), we have
where {R}_{d+1}\left(z\right)=\left(l\phantom{\rule{2.77695pt}{0ex}}qmk\right)\left(l\phantom{\rule{2.77695pt}{0ex}}qmk1\right)\cdots \left(l\phantom{\rule{2.77695pt}{0ex}}qmkd\right){z}^{\left(d+1\right)m\left(dl+1\right)}+{r}_{d+1}^{\left(\left(d+1\right)m\left(dl+1\right)1\right)}{z}^{\left(d+1\right)m\left(dl+1\right)1}+\cdots +{r}_{d+1}^{\left(1\right)}z+{r}_{d+1}^{\left(0\right)} is a polynomial, {r}_{d+1}^{\left(\left(d+1\right)m\left(dl+1\right)1\right)},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{r}_{d+1}^{\left(1\right)},\phantom{\rule{2.77695pt}{0ex}}{r}_{d+1}^{\left(0\right)} are constants.
By (2.2) and (2.4), since d ≥ l, then deg h_{ k } (z) + p  nk = q + km + d ≤ k(m + n  1) + p  nk, so p  q ≥ k + d. Observe the form of h_{ k } (z), then deg h_{ k } (z) = k(m + n  1), as deg h_{ k } (z) + p  nk = q + km + d, thus p  q = k + d. From (2.3) and (2.5), p  nk  nd  n ≤ (d + 1)m  (d  l + 1). Since pnkndn\left(d+1\right)m\ge p\frac{p\left(k+d+1\right)}{k+2d+2}\frac{q\left(d+1\right)}{k+2d+2}=\frac{\left(k+d\right)\left(d+1\right)}{k+2d+2}, then l  d ≥ p  nk  nd  n  (d + 1)m + 1 ≥ 1, it contradicts with d ≥ l.
Subcase 1.2. When d < l.
Differentiating both sides of (2.4), we have
where {U}_{d+1}\left(z\right)=D\left(l\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}qmk\right)\left(lqmk1\right)\cdots \left(l\phantom{\rule{2.77695pt}{0ex}}qmkd\right){z}^{\left(d+1\right)m}+\cdots +{u}_{d+1}^{\left(1\right)}z+{u}_{d+1}^{\left(0\right)} is a polynomial, {u}_{d+1}^{\left(i\right)}\left(i=0,\phantom{\rule{2.77695pt}{0ex}}1\right) are constants.
Differentiating both sides of (2.4) step by step for d times, we can get that ξ_{0} is a zero of g^{(k+d)}(z)  p^{(d)}(z), as p^{(d)}(z) ≠ 0, then γ_{ i } ≠ ξ_{0}(i = 1, 2, ..., n).
Here, we discuss in three subcases.
Subcase 1.2.1. When l < q + km + d. Similar to the proof of Subcase 1.1, we get p  q = k + d. By (2.3), (2.6), and γ_{ i } ≠ ξ_{0}(i = 1, 2, ..., n), (d + 1)m ≥ p  nk  nd  n, then
which is a contradiction.
Subcase 1.2.2. When l = q + km + d. If p > q, from (2.3), (2.6), and γ_{ i } ≠ ξ_{0}(i = 1, 2, ..., n), we get (d + 1)m ≥ p  nk  nd  n, then p\le nk+nd+n+\left(d+1\right)m\le \frac{p\left(k+d+1\right)}{k+2d+2}+\frac{q\left(d+1\right)}{k+2d+2}<p, which is a contradiction. Thus, p ≤ q. Then, from (2.3), (2.6), and γ_{ i } ≠ ξ_{0}(i = 1, 2, ..., n), we have l  d  1 ≤ (k + d + 1)(n + m  1). By simple calculation,
which is a contradiction.
Subcase 1.2.3. When l > q + km + d. If p ≤ q, by (2.2) and (2.4), then l ≤ q + km + d, which is a contradiction. Thus, p > q. By (2.3), (2.6) and γ_{ i } ≠ ξ_{0}(i = 1, 2, ..., n), we get (d + 1)m ≥ p  nk  nd  n, then p\le nk+nd+n+\left(d+1\right)m\le \frac{p\left(k+d+1\right)}{k+2d+2}+\frac{q\left(d+1\right)}{k+2d+2}<p, which is a contradiction.
Case 2. If g^{(k)}(z)  p(z) has no zero. Then, l = 0 in (2.4), similar discussion to Case 1, we get a contradiction.
By Case 1 and Case 2, g^{(k)}(z)  p(z) has at least two distinct zeros.
Lemma 2.3 (see [11]). Let g(z) be a transcendental meromorphic function of finite order on C, and let p(z) be a polynomial, p(z) ≢ 0. Suppose that all zeros of g(z) have multiplicity at least k + 1. Then, g^{(k)}(z)  p(z) has infinitely many zeros.
Lemma 2.4 (see [12]). Let \mathcal{F} be a family of meromorphic functions on the unit disk Δ, such that all zeros of functions in \mathcal{F} have multiplicity greater than or equal to l, and all poles of functions in \mathcal{F} have multiplicity greater than or equal to j. Let α be a real number satisfying l < α < j. Then, \mathcal{F} is not normal in any neighborhood of z_{0} ∈ Δ, if and only if there exist

(i)
points z_{ k } ∈ Δ, z_{ k } → z _{0};

(ii)
positive numbers ρ_{ k } , ρ_{ k } → 0, and

(iii)
functions {f}_{k}\in \mathcal{F} such that {\rho}_{k}^{\alpha}{f}_{k}\left({z}_{k}+{\rho}_{k}\zeta \right)\to g\left(\zeta \right) spherically locally uniformly in C, where g(ζ) is a nonconstant meromorphic function. The function may be taken to satisfy the normalization g ^{#}(z) ≤ g ^{#}(0) = 1(z ∈ C). Here, g ^{#}(z) denotes the spherical derivative of g(z).
Lemma 2.5 (see [13]). Suppose that f(z) is a meromorphic function on C, if the spherical derivative f^{#}(z) of f(z) is bounded, then the order of f(z) is at most 2.
Lemma 2.6 (see [14]). Suppose that f(z) is a transcendental meromorphic on C, k is a positive integer. If ε is a positive number, a, b are two dinstinct finite complex numbers, then
Lemma 2.7 (see [15]). Let g be a meromorphic function with finite order on C. If g has only finitely many critical values, then it has only finitely many asymptotic values.
Lemma 2.8 (see [16]). Let f(z) be meromorphic on C such that the set of its finite critical and asymptotic values is bounded. Then, there exists a positive number r_{0}, such that if z > r_{0} and f(z) > r_{0}, then
Lemma 2.9. Suppose that f(z) is a meromorphic function of finite order on C, a and b are two dinstinct finite complex numbers. If f'(z) = a ⇒ f(z) = 0, and f'(z) ≠ b, then f is a rational function.
Proof Suppose that f is not a rational function. Since f is a meromorphic function of finite order, then f' is also a meromorphic function of finite order. Because f'(z) ≠ b, then from Lemma 2.6, f'  a has infinitely many zeros {z_{ n } }, and z_{ n } → ∞(n → ∞). Since f'(z) = a ⇒ f(z) = 0, hence f(z_{ n } ) = 0.
Let F(z) = f(z)  bz, then F'(z) = f'(z)  b ≠ 0, i.e., F has no finite critical value. By Lemmas 2.7 and 2.8, we have
By simple calculation, we have
which is a contradiction. Then, f is a rational function.
Lemma 2.10 (see [6]). Let f\left(z\right)={a}_{n}{z}^{n}+{a}_{n1}{z}^{n1}+\cdots +{a}_{0}+\frac{q\left(z\right)}{p\left(z\right)}, where a_{0}, a_{1}, ..., a_{ n } are constants with a_{ n } ≠ 0, q(z) and p(z) are two coprime polynomials with deg q(z) < deg p(z), k be a positive integer. If f^{(k)}≠ 1, then we have

(i)
n = k, and k!a_{ k } = 1;

(ii)
f\left(z\right)=\frac{1}{k!}{z}^{k}+\cdots +{a}_{0}+\frac{1}{{\left(az+b\right)}^{m}};

(iii)
If the zeros of f are of order ≥ k + 1, then m = 1 in (ii) and f\left(z\right)=\frac{{\left(cz+d\right)}^{k+1}}{az+b}, where c(≠ 0), d are constants.
Lemma 2.11 (see [6, 17]). Let R be a rational function such that R' ≠ 0 on C. Then, either R = az + b or R=\frac{a}{{\left(z+{z}_{0}\right)}^{n}}+b, where a(≠ 0), b and z_{0} are constants, and n is a positive integer.
3. Proof of Theorem 1.1
For any point z_{0} in D, either p(z_{0}) = 0 or p(z_{0}) ≠ 0.
Case 1. When p(z_{0}) = 0. We may assume z_{0} = 0. Then, p(z) = a_{ l }z^{l} + a_{l+1}z^{l+1}+ · · · = z^{l}h(z), where a_{ l } (≠ 0), a_{l+1}, · · · are constants, l ≥ 1.
Let M=\left\{{G}_{j}{G}_{j}=\frac{{f}_{j}\left(z\right)}{{z}^{l}},\phantom{\rule{2.77695pt}{0ex}}{f}_{j}\in \mathcal{F}\right\}. If M is not normal at 0, then by Lemma 2.4, there exist points z_{ t } → 0, positive numbers ρ_{ t } → 0 and G_{ t } ∈ M such that {g}_{t}\left(\zeta \right)={\rho}_{t}^{k}{G}_{t}\left({z}_{t}+{\rho}_{t}\zeta \right)\to g\left(\zeta \right) spherically locally uniformly in C, where g(ζ) is a nonconstant meromorphic function in C, and g^{#}(ζ) ≤ 1. Thus, the order of g(ζ) is at most 2.
Here, we discuss in two cases.
Case 1.1. There exists a subsequence of \frac{{z}_{t}}{{\rho}_{t}}; we may still denote it as \frac{{z}_{t}}{{\rho}_{t}} such that \frac{{z}_{t}}{{\rho}_{t}}\to c, c is a finite complex number. Then,
spherically locally uniformly in C, so
spherically locally uniformly in C.
Since \forall f\in \mathcal{F}, the multiplicity of all zeros of f is at least k + d + 2, the multiplicity of all zeros of H is at least k + d + 2, then from Lemmas 2.1 and 2.3 H^{(k)}(ζ)  a_{ l }ζ^{l} ≢ 0, and H^{(k)}(ζ)  a_{ l }ζ^{l} has at least two distinct zeros.
Suppose that ζ_{1}, ζ_{2} are two distinct zeros of H^{(k)}(ζ)  a_{ l }ζ^{l} . We choose proper σ > 0, such that D(ζ_{1}, σ) ∩ D(ζ_{2}, σ) = ∅, where D(ζ_{1}, σ) = {ζ  ζ  ζ_{1} < σ}, D(ζ_{2}, σ) = {ζ  ζ  ζ_{2} < σ}.
By Hurwitz's Theorem, there exists a subsequence of {f}_{t}^{\left(k\right)}\left({\rho}_{t}\zeta \right)p\left({\rho}_{t}\zeta \right), we may still denote it as {f}_{t}^{\left(k\right)}\left({\rho}_{t}\zeta \right)p\left({\rho}_{t}\zeta \right), then exists points ζ_{ t } = ζ_{1}, and points {\stackrel{\u0303}{\zeta}}_{t}={\zeta}_{2}, such that when t is large enough,
As for each pair of functions f and g in \mathcal{F}, f^{(k)}and g^{(k)}share p(z) in D, then by (3.1) we can deduce that for any positive integer r, when t is large enough,
For fixed r, let t converges to ∞ in (3.2), then ρ_{ t }ζ_{ t } → 0, {\rho}_{t}{\stackrel{\u0303}{\zeta}}_{t}\to 0, thus
Then, by the isolation property of zeros of {f}_{r}^{\left(k\right)}\left(\zeta \right)p\left(\zeta \right), when t is large enough,
Thus, when t is large enough, ζ_{1} = ζ_{2} = 0, which contradicts with D(ζ_{1}, σ) ∩ (ζ_{2}, σ) = ∅. Thus, M is normal at 0.
Case 1.2. There exists a subsequence of \frac{{z}_{t}}{{\rho}_{t}}; we may still denote it as \frac{{z}_{t}}{{\rho}_{t}} such that \frac{{z}_{t}}{{\rho}_{t}}\to \infty. Then,
where {c}_{i}=l\left(l1\right)\cdots \left(li+1\right){C}_{k}^{l} when l ≥ i, and c_{ i } = 0 when l < i.
Thus, we have
spherically locally uniformly in C  {ζ  g(ζ) = ∞}.
Since \forall f\in \mathcal{F}, the multiplicity of all zeros of f is at least k + d + 2, the multiplicity of all zeros of g is at least k + 2. Thus, by Lemmas 2.1 and 2.3, g^{(k)}(ζ)  a_{ l } has at least two distinct zeros, and g^{(k)}(ζ)  a_{ l } ≢ 0. Suppose that ζ_{3}, ζ_{4} are two distinct zeros of g^{(k)}(ζ)  a_{ l } . We choose proper σ > 0, such that D(ζ_{3}, σ) ∩ D(ζ_{4}, σ) = ∅, where D(ζ_{3}, σ) = {ζ  ζ  ζ_{3} < σ}, D(ζ_{4}, σ) = {ζ  ζ  ζ_{4} < σ}.
By Hurwitz's Theorem, there exists a subsequence of {a}_{l}{f}_{t}^{\left(k\right)}\left({z}_{t}+{\rho}_{t}\zeta \right){a}_{l}p\left({z}_{t}+{\rho}_{t}\zeta \right), we may still denote it as {a}_{l}{f}_{t}^{\left(k\right)}\left({z}_{t}+{\rho}_{t}\zeta \right){a}_{l}p\left({z}_{t}+{\rho}_{t}\zeta \right), then exists points \hat{{\zeta}_{t}}={\zeta}_{3}, and points {\zeta}_{t}^{*}={\zeta}_{4}, such that when t is large enough,
Similar to the proof of Case 1.1, we get a contradiction. Then, M is normal at 0.
From Cases 1.1 and 1.2, we know M is normal at 0; there exists Δ _{ ρ } = {z : z < ρ } and a subsequence {G}_{{j}_{k}} of G_{ j } , such that {G}_{{j}_{k}} converges spherically locally uniformly to a meromorphic function G(z) or ∞(k → ∞) in Δ_{ ρ }.
Here, we discuss in two cases:
Case i. When k is large enough, {f}_{{j}_{k}}\left(0\right)\ne 0. Then, G(0) = ∞. Thus, for ∀ constant R > 0, ∃σ ∈ (0, ρ), we have G(z) > R when z ∈ Δ_{ σ }. Thus, for sufficiently large k,\phantom{\rule{2.77695pt}{0ex}}\left{G}_{{j}_{k}}\left(z\right)\right\phantom{\rule{2.77695pt}{0ex}}>\frac{R}{2},\phantom{\rule{2.77695pt}{0ex}}\frac{1}{{f}_{{j}_{k}}} is holomorphic in Δ_{ σ }. Hence when \leftz\right\phantom{\rule{2.77695pt}{0ex}}=\frac{\sigma}{2},
By Maximum Principle and Montel's Theorem, \mathcal{F} is normal at z = 0.
Case ii. There exists a subsequence of {f}_{{j}_{k}}; we may still denote it as {f}_{{j}_{k}} such that {f}_{{j}_{k}}\left(0\right)=0. Since \forall f\in \mathcal{F}, the multiplicity of all zeros of f is at least k + d + 2, then G(0) = 0. Thus, there exists 0 < r < ρ such that G(z) is holomorphic in Δ _{ r } = {z : z < r} and has a unique zero z = 0 in Δ_{ r }. Then, {G}_{{j}_{k}} converges spherically locally uniformly to a holomorphic function G(z) in Δ_{ r }; {f}_{{j}_{k}} converges spherically locally uniformly to a holomorphic function z^{l}G(z) in Δ_{ r }. Hence \mathcal{F} is normal at z = 0.
By Case i and Case ii, \mathcal{F} is normal at 0.
Case 2. When p(z_{0}) ≠ 0.
Suppose that \mathcal{F} is not normal at z_{0}. By Lemma 2.4, there exist points z_{ n } → z_{0}, ρ_{ n } → 0, {f}_{n}\in \mathcal{F} such that {\rho}_{n}^{k}{f}_{n}\left({z}_{n}+{\rho}_{n}\zeta \right)\to g\left(\zeta \right) spherically locally uniformly in C, g(ζ) is a nonconstant meromorphic function in C, and g^{#}(ζ) ≤ 1.
By Lemma 2.5, g(ζ) is a meromorphic function of finite order, the multiplicity of all zeros of f is at least k + d + 2 for \forall f\in \mathcal{F}, thus the multiplicity of all zeros of g is at least k + d + 2.
Hence by Lemmas 2.1 and 2.3, g^{(k)}(ζ)  p(z_{0}) has at least two distinct zeros, and g^{(k)}(ζ)  p(z_{0}) ≢ 0. Similar to the proof of Case 1.1, we get a contradiction. Thus, \mathcal{F} is normal at z_{0}.
Thus, \mathcal{F} is normal in D as z_{0} is arbitrary. The proof is complete.
4. Proof of Theorem 1.2
Similar to the proof of Theorem 1.1 and utilize Lemma 2.2, we can prove Theorem 1.2 immediately.
5. Proof of Theorem 1.3
Suppose that \mathcal{F} is not normal in D, then there exists z_{0} ∈ D, such that \mathcal{F} is not normal at z_{0}. Without loss of generality, we may assume z_{0} = 0. By Lemma 2.4, there exist z_{ k } → z_{0}, ρ_{ k } → 0, {f}_{k}\in \mathcal{F} such that {\rho}_{k}^{1}{f}_{k}\left({z}_{k}+{\rho}_{k}z\right)\to R\left(z\right) spherically locally uniformly in C, R(z) is a nonconstant meromorphic function in C, and R^{#}(z) ≤ 1. Thus, {f}_{k}^{\prime}\left({z}_{k}+{\rho}_{k}z\right)\to {R}^{\prime}\left(z\right) spherically locally uniformly in C.
By Hurwitz's Theorem, we can easily get: (i) R' = a ⇒ R = 0; (ii) R' ≠ b; (iii) the multiplicity of zeros of R is at least 2.
By Lemma 2.5, R(z) is a meromorphic function of finite order. Moreover from (i), (ii), and Lemma 2.9, R is a rational function.
Here, we discuss in two cases:
Case 1. When b = 0. Because R' ≠ b, by Lemma 2.11, R = cz + d or R=\frac{c}{{\left(z+{z}_{0}\right)}^{n}}+d, where c(≠ 0), d, z_{0} are constants, and n is a positive integer. From (iii), we have R=\frac{c}{{\left(z+{z}_{0}\right)}^{n}}+d. If a ≠ 0, then the roots number of {R}^{\prime}=\frac{nc}{{\left(z+{z}_{0}\right)}^{n+1}}=a is n + 1, and the roots are different from each other, meanwhile the roots number of R = 0 is at most n, it contradicts with R' = a ⇒ R = 0. Then, a = 0, which contradicts with that a and b are distinct constants.
Case 2. When b ≠ 0.
Here, we discuss in three subcases:
Subcase 2.1. If R is a polynomial. From R' ≠ b and Lemma 2.11, we have R = cz + d, where c(≠ 0), d(≠ b) are constants, which contradicts with (iii).
Subcase 2.2. If R=\frac{q\left(z\right)}{p\left(z\right)}, where q(z) and p(z) are coprime polynomials, deg q(z) < deg p(z), then {R}^{\prime}=\frac{{q}^{\prime}pq{p}^{\prime}}{{p}^{2}}. Since deg q(z) < deg p(z), so deg (q' p  qp') < deg (p^{2}). Thus, 0 is the only one deficiency value of R', which contradicts with R' ≠ b, b ≠ 0.
Subcase 2.3. If R={a}_{n}{z}^{n}+{a}_{n1}{z}^{n1}+\cdots +{a}_{0}+\frac{q\left(z\right)}{p\left(z\right)}, where a_{0}, a_{1}, ..., a_{ n } are constants, a_{ n } ≠ 0, q(z) and p(z) are coprime polynomials, deg q(z) < deg p(z). From Lemma 2.10 and (iii), R=bz+{a}_{0}+\frac{1}{cz+d}=\frac{{\left(lz+t\right)}^{2}}{cz+d}, where c(≠ 0), d, l(≠ 0), t are constants. Since a and b are distinct, then the roots of {R}^{\prime}a=ba\frac{c}{{\left(cz+d\right)}^{2}}=0 are two distinct complex numbers, meanwhile R = 0 has a single root z=\frac{t}{l}, which contradicts with R' = a ⇒ R = 0.
Then, \mathcal{F} is normal in D. The proof is complete.
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Acknowledgements
The authors would like to thank the referees for their detailed and valuable comments. The work was supported by the National Natural Science Foundation of China(Grant No. 11171013) and the Fundamental Research Funds for the Central Universities.
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Jiang, Y., Gao, Z. Normal families of meromorphic functions sharing values or functions. J Inequal Appl 2011, 72 (2011). https://doi.org/10.1186/1029242X201172
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DOI: https://doi.org/10.1186/1029242X201172