Normality of meromorphic functions and differential polynomials share values

Abstract

In this paper, we discuss the normality of meromorphic functions which involves differential polynomial sharing values. We obtain two results: Let k be a positive integer, b (≠0) be a complex number, and $h(z)$ be a polynomial with degree at least 2, and $H(f,f^{\prime },\dots ,f^{(k)})$ be a differential polynomial with $\frac{\mathrm{\Gamma }}{\gamma }{|}_H<k+1$. Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least $k+1$. If $h(z)-1$ has at least two distinct zeros, $h(f^{(k)})+H(f,f^{\prime },\dots ,f^{(k)})-1$ has at most one distinct zero in D for each $f\in \mathcal{F}$, then ℱ is normal in D. If $h(z)-b$ has at least two distinct zeros and for each pair of functions f and g in ℱ, $h(f^{(k)})+H(f,f^{\prime },\dots ,f^{(k)})$ and $h(g^{(k)})+H(g,g^{\prime },\dots ,g^{(k)})$ share b in D, then ℱ is normal in D, too. Two examples show that a condition in our results is necessary and our results improve Fang and Hong’s, and Zeng’s corresponding results.

MSC:30D35, 34A05.

1 Introduction and main results

Let D be a domain in ℂ, and ℱ be a family of meromorphic functions defined in the domain D. ℱ is said to be normal in D, in the sense of Montel, if for every sequence $\{f_n\}\subseteq \mathcal{F}$ contains a subsequence $\{f_{n_j}\}$ such that $f_{n_j}$ converges spherically uniformly on compact subsets of D.

ℱ is said to be normal at a point $z_0\in D$ if there exists a neighborhood of $z_0$ in which ℱ is normal. It is well known that ℱ is normal in a domain D if and only if it is normal at each of its points.

Let f and g be meromorphic functions defined in a domain D, and a and b be complex numbers. If $g(z)=b$ whenever $f(z)=a$, we write $f(z)=a\Rightarrow g(z)=b$. If $f(z)=a\Rightarrow g(z)=b$ and $g(z)=b\Rightarrow f(z)=a$, we write $f(z)=a⟺g(z)=b$; If $f(z)=a⟺g(z)=a$, we say that f and g share the value a in D.

Let $n_0,n_1,\dots ,n_k$ be non-negative integers and one of them nonzero at least, and set

$$\begin{array}{c}M(f,f^{\prime },\dots ,f^{(k)})=f^{n_0}{\left(f^{\prime }\right)}^{n_1}\cdots {\left(f^{(k)}\right)}^{n_k},\hfill \\ {\gamma }_M=n_0+n_1+n_2+\cdots +n_k,\hfill \\ {\mathrm{\Gamma }}_M=n_0+2n_1+3n_2+\cdots +(k+1)n_k.\hfill \end{array}$$

$M(f,f^{\prime },\dots ,f^{(k)})$ is called the differential monomial of f, ${\gamma }_M$ the degree of $M(f,f^{\prime },\dots ,f^{(k)})$ and ${\mathrm{\Gamma }}_M$ the weight of $M(f,f^{\prime },\dots ,f^{(k)})$.

Let $M_1(f,f^{\prime },\dots ,f^{(k)}),M_2(f,f^{\prime },\dots ,f^{(k)}),\dots ,M_m(f,f^{\prime },\dots ,f^{(k)})$ be differential monomials of f, and let $a_1(z),a_2(z),\dots ,a_m(z)$ be analytic in D. Set

$$\begin{array}{c}H(f,f^{\prime },\dots ,f^{(k)})=a_1(z)M_1(f,f^{\prime },\dots ,f^{(k)})+\cdots +a_m(z)M_m(f,f^{\prime },\dots ,f^{(k)}),\hfill \\ {\gamma }_H=max\{{\gamma }_{M_1},{\gamma }_{M_2},\dots ,{\gamma }_{M_m}\},\hfill \\ {\mathrm{\Gamma }}_H=max\{{\mathrm{\Gamma }}_{M_1},{\mathrm{\Gamma }}_{M_2},\dots ,{\mathrm{\Gamma }}_{M_m}\}.\hfill \end{array}$$

$H(f,f^{\prime },\dots ,f^{(k)})$ is called a differential polynomial of f, ${\gamma }_H$ the degree of $H(f,f^{\prime },\dots ,f^{(k)})$ and ${\mathrm{\Gamma }}_H$ the weight of $H(f,f^{\prime },\dots ,f^{(k)})$. If ${\gamma }_{M_1}={\gamma }_{M_2}=\cdots ={\gamma }_{M_m}=t$, then $H(f,f^{\prime },\dots ,f^{(k)})$ is called a homogeneous differential polynomial of degree t. Set

$$\frac{\mathrm{\Gamma }}{\gamma }{|}_H=max\{\frac{{\mathrm{\Gamma }}_{M_1}}{{\gamma }_{M_1}},\frac{{\mathrm{\Gamma }}_{M_2}}{{\gamma }_{M_2}},\dots ,\frac{{\mathrm{\Gamma }}_{M_m}}{{\gamma }_{M_m}}\}.$$

The following theorem was proved by Fang and Hong [1].

Theorem 1.1 [1]

Let ℱ be a family of meromorphic functions defined in D, k and q (≥2) be two positive integers, and $H(f,f^{\prime },\dots ,f^{(k)})$ be a differential polynomial with $\frac{\mathrm{\Gamma }}{\gamma }{|}_H<k+1$. If the zeros of $f(z)$ are of multiplicity at least $k+1$ and ${(f^{(k)})}^q+H(f,f^{\prime },\dots ,f^{(k)})\ne 1$ for each $f\in \mathcal{F}$, then ℱ is normal in D.

It is natural to ask whether the condition in Theorem 1.1 that ${(f^{(k)})}^q+H(f,f^{\prime },\dots ,f^{(k)})\ne 1$ can be relaxed. In this paper we investigate this problem and prove the following result.

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 $H(f,f^{\prime },\dots ,f^{(k)})$ be a differential polynomial with $\frac{\mathrm{\Gamma }}{\gamma }{|}_H<k+1$. If $h(z)-1$ has at least two distinct zeros, the zeros of $f(z)$ are of multiplicity at least $k+1$ and $h(f^{(k)})+H(f,f^{\prime },\dots ,f^{(k)})-1$ has at most one distinct zero in D for each $f\in \mathcal{F}$, then ℱ is normal in D.

By the idea of shared values, very recently, Zeng [2] proved the following theorem.

Theorem 1.3 [2]

Let k and q (≥2) be two positive integers, $b\ne 0$ be a complex number, and let $H(f,f^{\prime },\dots ,f^{(k)})$ be a differential polynomial with $\frac{\mathrm{\Gamma }}{\gamma }{|}_H<k+1$. Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least $k+1$. If for each pair of functions f and g in ℱ, ${(f^{(k)})}^q+H(f,f^{\prime },\dots ,f^{(k)})$ and ${(g^{(k)})}^q+H(g,g^{\prime },\dots ,g^{(k)})$ share b in D, then ℱ is normal in D.

It is natural to ask whether Theorem 1.3 can be improved. In this paper, we study this problem and obtain the following theorem.

Theorem 1.4 Let k be a positive integer, b (≠0) be a complex number, $h(z)$ be a polynomial, and let $H(f,f^{\prime },\dots ,f^{(k)})$ be a differential polynomial with $\frac{\mathrm{\Gamma }}{\gamma }{|}_H<k+1$. Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least $k+1$. If $h(z)-b$ has at least two distinct zeros and for each pair of functions f and g in ℱ, $h(f^{(k)})+H(f,f^{\prime },\dots ,f^{(k)})$ and $h(g^{(k)})+H(g,g^{\prime },\dots ,g^{(k)})$ share b in D, then ℱ is normal in D.

Example 1.1 Let $D=\{z:|z|<1\}$, $h(z)=z^{k+1}+1$ and $\mathcal{F}:=\{f_n(z)=n{z}^{k+1}\}$. Then

$$h(f_n^{(k)}(z))+f_n(z)=n[n^k{((k+1)!)}^{k+1}+1]z^{k+1}+1.$$

We can see that $\frac{\mathrm{\Gamma }}{\gamma }{|}_H=1<k+1$, $h(f_n^{(k)}(z))+f_n(z)-1$ has only one distinct zero in D for each function $f_n$ in ℱ, and $h(f_n^{(k)}(z))+f_n(z)$ and $h(f_m^{(k)}(z))+f_m(z)$ share 1 in D for each pair of functions $f_n$ and $f_m$ in ℱ. On the other hand, $f_j(0)=0$, $f_j(\frac{1}{j^{\frac{1}{k+1}}})=1$, for any $j\in \mathbb{N}$. This implies that the family ℱ fails to be equicontinuous at 0, and thus ℱ is not normal at 0.

Remark 1.5 This example shows that $h(z)-1$ to have at least two distinct zeros ($h(z)-b$ to have at least two distinct zeros) is necessary in Theorem 1.2 (Theorem 1.4).

Example 1.2 Let $D=\{z:|z|<1\}$, $h(z)=z^{k+1}+z^k+1$ and $\mathcal{F}:=\{f_n(z)=z^{k+1}\}$. Then

$$h(f_n^{(k)}(z))-{[(k+1)!]}^{k+1}{f}_n(z)+f_n^{\prime }(z)=[{((k+1)!)}^k+k+1]z^k+1.$$

We can see that $\frac{\mathrm{\Gamma }}{\gamma }{|}_H=2<k+1$ if $k\ge 2$, and for each pair of functions $f_n$ and $f_m$ in ℱ, $h(f_n^{(k)}(z))-{[(k+1)!]}^{k+1}{f}_n(z)+f_n^{\prime }(z)$ and $h(f_m^{(k)}(z))-{[(k+1)!]}^{k+1}{f}_m(z)+f_m^{\prime }(z)$ share 1. Therefore, ℱ is normal in D by our Theorem 1.4.

Remark 1.6 From this example we also know that $h(f_n^{(k)}(z))-{[(k+1)!]}^{k+1}{f}_n(z)+f_n^{\prime }(z)-1$ has only one solution in D for each $f_n$ in ℱ. The case of shared b includes the case of ≠b, that is to say, Theorem 1.2 is a generalization of Theorem 1.1 and Theorem 1.4 is a generalization of Theorem 1.3.

2 Preliminary lemmas

In order to prove our results, we need the following lemmas. The first one is Zalcman’s Theorem.

Lemma 2.1 [3]

Let $k\in {\mathbf{N}}_{+}$, let ℱ be a family of functions meromorphic on the unit disc  Δ, all of whose zeros have multiplicity at least k, and suppose that there exists $A\ge 1$ such that $|f^{(k)}(z)|\le A$ whenever $f(z)=0$. Then if ℱ is not normal at $z_0$, there exist, for each $0\le \alpha \le k$,

• (a) functions $f_n\in \mathcal{F}$;

• (b) points $z_n\in \mathrm{\Delta }$, $z_n\to z_0$, and

• (c) positive numbers ${\rho }_n\to 0^{+}$

such that $g_n(\zeta )={\rho }_n^{-\alpha }{f}_n(z_n+{\rho }_n\zeta )\to g(\zeta )$ locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on ℂ, all of whose zeros have multiplicity at least k, such that $g^{\mathrm{\#}}(\zeta )\le g^{\mathrm{\#}}(0)=kA+1$. In particular, g has order at most 2.

Here $g^{\mathrm{\#}}(z)$ denotes the spherical derivative

$$g^{\mathrm{\#}}(z)=\frac{|g^{\prime }(z)|}{1+{|g(z)|}^2}.$$

Lemma 2.2 [4]

Let $f(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_0+q(z)/p(z)$, where $a_0,a_1,\dots ,a_n$ are constants with $a_n\ne 0$, and q and p are two co-prime polynomials, neither of which vanishes identically, with $degq<degp$, and let k be a positive integer and b a nonzero complex number. If $f^{(k)}\ne b$, and the zeros of f all have multiplicity at least $k+1$, then

$$f(z)=\frac{b{(z-d)}^{k+1}}{k!(z-c)},$$

where c and d are distinct complex numbers.

Lemma 2.3 Let g be a nonconstant meromorphic function, and $h(z)$ be a polynomial. If $h(z)$ has at least two distinct zeros and all zeros of g have multiplicity at least $k+1$, then $h(g^{(k)}(\xi ))$ has at least two distinct zeros.

Proof Case 1. If $h(g^{(k)}(\xi ))$ has only one zero α, then $h(g^{(k)}(\alpha ))=0$, and $\xi \ne \alpha$, $h(g^{(k)}(\xi ))\ne 0$.

Suppose that $d_i$ ($i=1,2$) are two distinct zeros of $h(z)$. Without loss of generality, we may assume that $g^{(k)}(\alpha )=d_1$, then $g^{(k)}(\xi )\ne d_1$ for $\xi \ne \alpha$.

Firstly, we will show that $g(\xi )$ is not a transcendental meromorphic function. By Nevanlinna Theory, we have

$$\begin{array}{rl}T(r,g^{(k)})& \le \overline{N}(r,g^{(k)})+\overline{N}(r,\frac{1}{g^{(k)}-d_1})+\overline{N}(r,\frac{1}{g^{(k)}-d_2})+S(r,g^{(k)})\\ \le \frac{1}{k+1}N(r,g^{(k)})+O(logr)+S(r,g^{(k)})\\ \le \frac{1}{k+1}T(r,g^{(k)})+O(logr)+S(r,g^{(k)}).\end{array}$$

Hence, we get $T(r,g^{(k)})=O(logr)+S(r,g^{(k)})$, it follows that $g(\xi )$ is not a transcendental meromorphic function.

If $g(\xi )$ is a polynomial, then

$$\begin{array}{rl}h(g^{(k)}(\xi ))& =h[d_1+c{(\xi -\alpha )}^n]\\ ={(\xi -\alpha )}^{n}Q(\xi ),\end{array}$$

where Q is a polynomial such that $Q(\alpha )\ne 0$ and the degree of Q is not less than 1. Thus there exists an ${\alpha }_1\ne \alpha$, such that $Q({\alpha }_1)=0$. That is to say, there exists an ${\alpha }_1\ne \alpha$, such that $h(g^{(k)}({\alpha }_1))=0$, which is a contradiction.

Therefore $g(\xi )$ is rational but not a polynomial. Under the conditions of Lemma 2.2 on the rational functions g, we have

$$g(\xi )=\frac{d_2{(\xi -d)}^{k+1}}{k!(\xi -c)},$$

where c and d are distinct complex numbers, $d_2\ne 0$, and then

$$g^{(k)}=d_2+\frac{A}{{(\xi -c)}^{k+1}},$$

where $A\ne 0$ is a complex number.

Hence $g^{(k)}(\xi )=d_1$ has $k+1$ distinct zeros, which contradicts $g^{(k)}(\xi )=d_1$ having only the zero $\xi =\alpha$.

Case 2. $h(g^{(k)}(\xi ))\ne 0$. Since $degh\ge 2$, by Nevanlinna Theory once more, we have

$$\begin{array}{rl}T(r,g^{(k)})& \le \overline{N}(r,g^{(k)})+\overline{N}(r,\frac{1}{g^{(k)}-d_1})+\overline{N}(r,\frac{1}{g^{(k)}-d_2})+S(r,g^{(k)})\\ \le \frac{1}{k+1}N(r,g^{(k)})+S(r,g^{(k)})\\ \le \frac{1}{k+1}T(r,g^{(k)})+S(r,g^{(k)}),\end{array}$$

where $d_i$ ($i=1,2$) are two distinct zeros of $h(z)$. Hence, we get $T(r,g^{(k)})=S(r,g^{(k)})$, and it follows that $g^{(k)}$ is a constant. This together with the fact that the zeros of g have multiplicity at least $k+1$ shows that g is a constant, a contradiction. □

3 Proofs of theorems

Proof of Theorem 1.2 We show that ℱ is normal in D. Otherwise, there exists at least one point $z_0\in D$ such that ℱ is not normal at $z_0$. Then by Lemma 2.1, we can find a subsequence of ℱ, which we may denote by $\{f_n\}$, $z_n\in \mathrm{\Delta }$, $z_n\to z_0$ and ${\rho }_n\to 0^{+}$ such that $g_n(\xi )={\rho }_n^{-k}{f}_n(z_n+{\rho }_n\xi )$ converges local uniformly with respect to the spherical metric to a nonconstant meromorphic function g on ℂ, all of whose zeros have multiplicity at least $k+1$.

It is easily seen that

$$\begin{array}{r}H(f_n,f_n^{\prime },\dots ,f_n^{(k)})(z_n+{\rho }_n\xi )\\ =\sum _{i=1}^m{a}_i(z_n+{\rho }_n\xi ){\rho }_n^{(k+1){\gamma }_{M_i}-{\mathrm{\Gamma }}_{M_i}}{M}_i(g_n,g_n^{\prime },\dots ,g_n^{(k)})(\xi ).\end{array}$$

Noting that all $a_i(z)$ ($i=1,2,\dots ,m$) are analytic on D implies

$$|a_i(z_n+{\rho }_n\xi )|\le M(\frac{1+r}{2},a_i(z))<\mathrm{\infty }$$

for sufficiently large n, we deduce from $\frac{\mathrm{\Gamma }}{\gamma }{|}_H<k+1$ that

$$\sum _{i=1}^m{a}_i(z_n+{\rho }_n\xi ){\rho }_n^{(k+1){\gamma }_{M_i}-{\mathrm{\Gamma }}_{M_i}}{M}_i(g_n,g_n^{\prime },\dots ,g_n^{(k)})(\xi )$$

converges uniformly to 0 on ℂ.

Thus we find that

$$h(g_n^{(k)}(\xi ))+\sum _{i=1}^m{a}_i(z_n+{\rho }_n\xi ){\rho }_n^{(k+1){\gamma }_{M_i}-{\mathrm{\Gamma }}_{M_i}}{M}_i(g_n,g_n^{\prime },\dots ,g_n^{(k)})(\xi )-1$$

converges local uniformly to $h(g^{(k)}(\xi ))-1$ on ℂ.

Hence, by Hurwitz’s Theorem, the hypothesis of the theorem, and Lemma 2.3, we see that $h(g^{(k)}(\xi ))\equiv 1$ or $h(g^{(k)}(\xi ))-1$ has at least two distinct zeros on ℂ.

Case 1. If $h(g^{(k)}(\xi ))\equiv 1$ on ℂ.

Then by $h(z)-1$ having at least two distinct zeros, we find that $h(z)-c$ has at least two distinct zeros except for at most one complex number c. Therefore Lemma 2.3 tells us that $h(g^{(k)}(\xi ))-c$ has zero for at least two distinct c except that $g(z)$ is a constant function. This is also impossible.

Case 2. If $h(g^{(k)}(\xi ))-1$ has at least two distinct zeros on ℂ.

Then, without out loss generality, let ${\xi }_0$ and ${\xi }_0^{\star }$ be two distinct zeros of $h(g^{(k)}(\xi ))-1$, and choose δ (>0) small enough such that $D({\xi }_0,\delta )\cap D({\xi }_0^{\star },\delta )=\mathrm{\varnothing }$, where $D({\xi }_0,\delta )=\{\xi :|\xi -{\xi }_0|<\delta \}$, and $D({\xi }_0^{\star },\delta )=\{\xi :|\xi -{\xi }_0^{\star }|<\delta \}$. By Hurwitz’s Theorem, there exist two sequences of points ${\xi }_n\to {\xi }_0$ and ${\xi }_n^{\star }\to {\xi }_0^{\star }$ such that for sufficiently large n

$$\begin{array}{c}h\left(f_n^{(k)}\right)(z_n+{\rho }_n{\xi }_n)+H(f_n,f_n^{\prime },\dots ,f_n^{(k)})(z_n+{\rho }_n{\xi }_n)-1=0,\hfill \\ h\left(f_n^{(k)}\right)(z_n+{\rho }_n{\xi }_n^{\star })+H(f_n,f_n^{\prime },\dots ,f_n^{(k)})(z_n+{\rho }_n{\xi }_n^{\star })-1=0.\hfill \end{array}$$

Hence, we have ${\xi }_n\in D({\xi }_0,\delta )$ and ${\xi }_n^{\star }\in D({\xi }_0^{\star },\delta )$ for sufficiently large n. Thus each $h(f_n^{(k)})(z)+H(f_n,f_n^{\prime },\dots ,f_n^{(k)})(z)-1$ has two distinct zeros for large enough n, which contradicts our hypothesis.

This contradiction shows that ℱ is normal in D and hence Theorem 1.2 is proved. □

Proof of Theorem 1.4 Suppose that ℱ is a family meromorphic and not normal in D. Then there exists at least one point $z_0\in D$ such that ℱ is not normal at the point $z_0$. By Lemma 2.1, there exist:

• (a) functions $f_n\in \mathcal{F}$;

• (b) points $z_n\in \mathrm{\Delta }$, $z_n\to z_0$, and

• (c) positive numbers ${\rho }_n\to 0^{+}$

such that $g_n(\zeta )={\rho }_n^{-k}{f}_n(z_n+{\rho }_n\zeta )\to g(\zeta )$ locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function on ℂ, all of whose zeros have multiplicity at least $k+1$.

It is easily seen that

$$\begin{array}{r}H(f_n,f_n^{\prime },\dots ,f_n^{(k)})(z_n+{\rho }_n\xi )\\ =\sum _{i=1}^m{a}_i(z_n+{\rho }_n\xi ){\rho }_n^{(k+1){\gamma }_{M_i}-{\mathrm{\Gamma }}_{M_i}}{M}_i(g_n,g_n^{\prime },\dots ,g_n^{(k)})(\xi ).\end{array}$$

Noting that all $a_i(z)$ ($i=1,2,\dots ,m$) are analytic on D implies

$$|a_i(z_n+{\rho }_n\xi )|\le M(\frac{1+r}{2},a_i(z))<\mathrm{\infty }$$

for sufficiently large n, we deduce from $\frac{\mathrm{\Gamma }}{\gamma }{|}_H<k+1$ that

$$\sum _{i=1}^m{a}_i(z_n+{\rho }_n\xi ){\rho }_n^{(k+1){\gamma }_{M_i}-{\mathrm{\Gamma }}_{M_i}}{M}_i(g_n,g_n^{\prime },\dots ,g_n^{(k)})(\xi )$$

converges uniformly to 0 on ℂ.

Thus we know that

$$h(g_n^{(k)}(\xi ))+\sum _{i=1}^m{a}_i(z_n+{\rho }_n\xi ){\rho }_n^{(k+1){\gamma }_{M_i}-{\mathrm{\Gamma }}_{M_i}}{M}_i(g_n,g_n^{\prime },\dots ,g_n^{(k)})(\xi )-b$$

converges local uniformly to $h(g^{(k)}(\xi ))-b$ on ℂ.

Take $f\in \{f_n\}$, we consider two cases.

Case 1. $[h(f^{(k)})+H(f,f^{\prime },\dots ,f^{(k)})](z_0)\ne b$.

Then there exists a positive number $\delta >0$ such that

$$[h\left(f_n^{(k)}\right)+H(f_n,f_n^{\prime },\dots ,f_n^{(k)})](z)\ne b$$

for all z in $D_{\delta }=\{z:|z-z_0|<\delta \}$, by sharing condition.

Hence, by Hurwitz’s Theorem, the hypothesis of the theorem, and Lemma 2.3, we see that $h(g^{(k)}(\xi ))\ne b$ or $h(g^{(k)}(\xi ))\equiv b$ on ℂ.

If $h(g^{(k)}(\xi ))\ne b$, then by Lemma 2.3 and the hypothesis of the theorem, we see that $h(g^{(k)}(\xi ))-b$ has at least two distinct zeros except that $g(z)$ is a constant function, a contradiction.

If $h(g^{(k)}(\xi ))\equiv b$, the same arguments of the proof of Case 1 in the proof of Theorem 1.3 implies that it does not hold.

Case 2. $[h(f^{(k)})+H(f,f^{\prime },\dots ,f^{(k)})](z_0)=b$.

Next we consider two subcases.

Subcase 2.1. $[h(f^{(k)})+H(f,f^{\prime },\dots ,f^{(k)})](z)\equiv b$ for all z in $D_{\delta }=\{z:|z-z_0|<\delta \}$. From the discussion above, we have $h(g^{(k)}(\xi ))\equiv b$ in ℂ. This is impossible.

Subcase 2.2. There exists a $\delta >0$ such that $h(f^{(k)})+H(f,f^{\prime },\dots ,f^{(k)})\ne b$ in $D_{\delta }^0=\{z:0<|z-z_0|<\delta \}$. By the supposition and discussion above, this means that

$$[h\left(f_n^{(k)}\right)+H(f_n,f_n^{\prime },\dots ,f_n^{(k)})](z_n+{\rho }_n\xi )\ne b$$

for $z_n+{\rho }_n\xi \ne z_0$, and

$$[h\left(f^{(k)}\right)+H(f,f^{\prime },\dots ,f^{(k)})](z_0)=b.$$

We claim that $h(g^{(k)}(\xi ))-b$ has just a unique zero.

Suppose that there exist two distinct zeros ${\xi }_0$ and ${\xi }_0^{\star }$, choose δ (>0) small enough such that $D({\xi }_0,\delta )\cap D({\xi }_0^{\star },\delta )=\mathrm{\varnothing }$, where $D({\xi }_0,\delta )=\{\xi :|\xi -{\xi }_0|<\delta \}$ and $D({\xi }_0^{\star },\delta )=\{\xi :|\xi -{\xi }_0^{\star }|<\delta \}$.

By Hurwitz’s Theorem, there exist points ${\xi }_n\in D({\xi }_0,\delta )$, ${\xi }_n^{\star }\in D({\xi }_0^{\star },\delta )$, such that for sufficiently large j

$$\begin{array}{c}h\left(f_n^{(k)}\right)(z_n+{\rho }_n{\xi }_n)+H(f_n,f_n^{\prime },\dots ,f_n^{(k)})(z_n+{\rho }_n{\xi }_n)-b=0,\hfill \\ h\left(f_n^{(k)}\right)(z_n+{\rho }_n{\xi }_n^{\star })+H(f_n,f_n^{\prime },\dots ,f_n^{(k)})(z_n+{\rho }_n{\xi }_n^{\star })-b=0.\hfill \end{array}$$

By the assumption that $h(f^{(k)})+H(f,f^{\prime },\dots ,f^{(k)})$ and $h(g^{(k)})+H(g,g^{\prime },\dots ,g^{(k)})$ share b in D for each pair of functions f and g in ℱ, we see that for any integer m

$$\begin{array}{r}h\left(f_m^{(k)}\right)(z_n+{\rho }_n{\xi }_n)+H(f_m,f_m^{\prime },\dots ,f_m^{(k)})(z_n+{\rho }_n{\xi }_n)-b=0,\\ h\left(f_m^{(k)}\right)(z_n+{\rho }_n{\xi }_n^{\star })+H(f_m,f_m^{\prime },\dots ,f_m^{(k)})(z_n+{\rho }_n{\xi }_n^{\star })-b=0.\end{array}$$

We fix m and note that $z_n+{\rho }_n{\xi }_n\to z_0$, $z_n+{\rho }_n{\xi }_n^{\star }\to z_0$ if $n\to \mathrm{\infty }$. From this we deduce

$$h\left(f_m^{(k)}\right)(z_0)+H(f_m,f_m^{\prime },\dots ,f_m^{(k)})(z_0)-b=0.$$

Since

$$[h\left(f_n^{(k)}\right)+H(f_n,f_n^{\prime },\dots ,f_n^{(k)})](z_n+{\rho }_n\xi )\ne b$$

if $z_n+{\rho }_n\xi \ne z_0$, and

$$[h\left(f^{(k)}\right)+H(f,f^{\prime },\dots ,f^{(k)})](z_0)=b,$$

noting that the zeros of

$$h\left(f_m^{(k)}\right)(z)+H(f_m,f_m^{\prime },\dots ,f_m^{(k)})(z)-b$$

have no accumulation point, for sufficiently large n, we have

$$z_n+{\rho }_n{\xi }_n=z_0,z_n+{\rho }_n{\xi }_n^{\star }=z_0.$$

Hence

$${\xi }_n=\frac{z_0-z_n}{{\rho }_n},{\xi }_n^{\star }=\frac{z_0-z_n}{{\rho }_n}.$$

This contradicts the fact that ${\xi }_n\in D({\xi }_0,\delta )$, ${\xi }_n^{\star }\in D({\xi }_0^{\star },\delta )$, and $D({\xi }_0,\delta )\cap D({\xi }_0^{\star },\delta )=\mathrm{\varnothing }$. So $h(g^{(k)}(\xi ))-b$ has just a unique zero ignoring multiplicity. This contradicts the conclusion of Lemma 2.3 that $h(g^{(k)}(\xi ))-b$ has at least two distinct zeros.

Hence ℱ is normal at $z_0$, and then ℱ is normal in D. The proof of Theorem 1.4 is complete. □