Normality Criteria for Families of Meromorphic Functions about Shared Functions

TL;DR

This paper establishes new normality criteria for families of meromorphic functions sharing analytic functions, extending previous results and broadening the understanding of function behavior in complex analysis.

Contribution

The paper introduces generalized normality criteria for meromorphic functions sharing analytic functions, expanding upon earlier work by Wang, Fang, Qui, and Zhu.

Findings

Established new normality criteria for meromorphic functions.

Extended previous results to broader classes of functions.

Provided conditions under which families of functions are normal.

Abstract

In this paper we prove some normality criteria for a family of meromorphic functions concerning shared analytic functions, which extend or generalized some result obtained by Y. F. Wang, M. L. Fang~\cite{WF} and J. Qui, T. Zhu ~\cite{QZ}.

Equations

$$E_f(p)=\{z\in D:f(z)=p\}$$

$$E_f(\infty )=\text{poles of }f\text{ in }D.$$

$$f^{(k)}(z)-p(z)=K(z-z_0{)}^l$$

$$f^{(k+d)}(z)-C=Kl(z-z_0{)}^{l-d},$$

$$f^{(k)}(z)=\frac{A(z-{\alpha }_1{)}^{m_1}(z-{\alpha }_2{)}^{m_2}\dots (z-{\alpha }_s{)}^{m_s}}{(z-{\beta }_1{)}^{n_1}(z-{\beta }_2{)}^{n_2}\dots (z-{\beta }_t{)}^{n_t}},$$

$$\sum _{i=1}^s{m}_i=M\ge (2d+2)s\text{and}\sum _{j=1}^t{n}_j=N\ge (k+1)t\ge (2d+3)t.$$

$$f^{(k+d+1)}(z)=\frac{(z-{\alpha }_1{)}^{m_1-d-1}(z-{\alpha }_2{)}^{m_2-d-1}\dots (z-{\alpha }_s{)}^{m_s-d-1}{g}_1(z)}{(z-{\beta }_1{)}^{n_1+d+1}(z-{\beta }_2{)}^{n_2+d+1}\dots (z-{\beta }_t)n_t+d+1},$$

$$f^{(k)}(z)=p(z)+\frac{B(z-z_0{)}^l}{(z-{\beta }_1{)}^{n_1}(z-{\beta }_2{)}^{n_2}\dots (z-{\beta }_t{)}^{n_t}}$$

$$f^{(k+d+1)}(z)=\frac{g_2(z)}{(z-{\beta }_1{)}^{n_1+d+1}(z-{\beta }_2{)}^{n_2+d+1}\dots (z-{\beta }_t{)}^{n_t+d+1}}$$

$$f^{(k+d+1)}(z)=\frac{(z-z_0{)}^{l-d-1}{g}_3(z)}{(z-{\beta }_1{)}^{n_1+d+1}(z-{\beta }_2{)}^{n_2+d+1}\dots (z-{\beta }_t{)}^{n_t+d+1}},$$

$$h(z)=z^l+a_{l+1}{z}^m+1+\dots =z^{l}b(z),$$

$${\mathcal{F}}_1:=\{F_n:F_n(z)=\frac{f_n(z)}{z^l},f\in \mathcal{F}\}.$$

$$g_n(\xi )={\rho }_n^{-k}{F}_n(z_n+{\rho }_n\xi )\to g(\xi )$$

$$G_n(\xi )=\frac{f_n({\rho }_n\xi )}{{\rho }_n^{k+l}}=({\rho }_n\xi {)}^l{F}_n(z_n+{\rho }_n(\xi -\frac{z_n}{{\rho }_n}))\to {\xi }^{l}g(\xi -\alpha )=G(\xi )$$

$$G_n^{(k)}(\xi )-\frac{h({\rho }_n\xi )}{{\rho }_n^l}=\frac{f_n^{(k)}({\rho }_n\xi )-h({\rho }_n\xi )}{{\rho }_n^l}\to G^{(k)}(\xi )-{\xi }^l$$

$$f_n^{(k)}({\rho }_n{\xi }_n)-h({\rho }_n{\xi }_n)=0,$$

$$f_n^{(k)}({\rho }_n{\xi }_n^{\ast })-h({\rho }_n{\xi }_n^{\ast })=0.$$

$$f_m^{(k)}({\rho }_n{\xi }_n)-h({\rho }_n{\xi }_n)=0,$$

$$f_m^{(k)}({\rho }_n{\xi }_n)-h({\rho }_n{\xi }_n^{\ast })=0.$$

$$f_m^{(k)}(0)-h(0)=0.$$

$${\rho }_n{\xi }_n={\rho }_n{\xi }_n^{\ast }=0.$$

$${\xi }_n={\xi }_n^{\ast }=0.$$

$$F_n^{(k)}(z)$$

$$=\frac{f_n^{(k)}(z)}{z^l}-\sum _{m=1}^k{C}_m\frac{F_n^{(k-m)}(z)}{z^m},$$

$$C_m=C_k^{m}l(l-1)\dots (l-m+1),\text{when}m\le l,\text{and}{C}_m=0,\text{when}m>l.$$

$$g_n^{(k)}(\xi )$$

$$=\frac{f_n^{(k)}(z_n+{\rho }_n\xi )}{(z_n+{\rho }_n\xi {)}^l}-\sum _{m=1}^k{C}_m{F}_n^{(k-m)}(z)\frac{1}{(z_n+{\rho }_n\xi {)}^m}$$

$$=\frac{f_n^{(k)}(z_n+{\rho }_n\xi )}{(z_n+{\rho }_n\xi {)}^l}-\sum _{m=1}^k{C}_m{g}_n^{(k-m)}(z)\frac{1}{(\frac{z_n}{{\rho }_n}+\xi {)}^m}.$$

$$\frac{f_n^{(k)}(z_n+{\rho }_n\xi )}{h(z_n+{\rho }_n\xi )}=\left(g_n^{(k)}(\xi )+\sum _{m=1}^k{g}_n^{(k-m)}(\xi )\frac{C_m}{(\frac{z_n}{{\rho }_n}+\xi {)}^m}\right)\frac{1}{b(z_n+{\rho }_n\xi )}.$$

$$\frac{f_n^{(k)}(z_n+{\rho }_n\xi )-h(z_n+{\rho }_n\xi )}{h(z_n+{\rho }_n\xi )}\to g^{(k)}(\xi )-1,$$

$$f_n^{(k)}(z_n+{\rho }_n\widehat{{\xi }_n})-h(z_n+{\rho }_n\widehat{{\xi }_n})=0.$$

$$f_n^{(k)}(z_n+{\rho }_n\widehat{{\xi }_n^{\ast }})-h(z_n+{\rho }_n\widehat{{\xi }_n^{\ast }})=0.$$

Full text

1. Introduction and main results

Let $D$ be a domain in $\mathbb{C}$, and $\mathcal{F}$ be a family of meromorphic function in a domain $D$. $\mathcal{F}$ is said to be normal in a domain $D$, in the sense of Montel, if for each sequence $\{f_{j}\}\in\mathcal{F}$ there exist a subsequence $\{f_{j_{k}}\}$, such that $\{f_{j_{k}}\}$ converges spherically locally uniformly on $D$, to a meromorphic function or $\infty$ [1, 4, 9, 12].

Wilhelm Schwick  [10] was the first who gave a connection between normality and shared values and proved a theorem which says that: A family $\mathcal{F}$ of meromorphic functions on a domain ${D}$ is normal, if $f$ and $f^{\prime}$ share $a_{1}$, $a_{2}$, $a_{3}$ for every $f\in\mathcal{F}$, where $a_{1}$, $a_{2}$, $a_{3}$ are distinct complex numbers.

Let us recall the definition of shared value. Let $f$ be a meromorphic function of a domain $D\subset\mathbb{C}$. For $p\in\mathbb{C}$, let

[TABLE]

and let

[TABLE]

For $p\in\mathbb{C}\cup\{\infty\}$, two meromorphic functions $f$ and $g$ of $D$ share the value $p$ if $E_{f}(p)=E_{g}(p).$

In 1998, Wang and Fang  [11] obtained the following result.

Theorem A**.**

*Let $\mathcal{F}$ be a family of meromorphic functions in a domain $D.$ Let $k$ be a positive integer and $b$ be a non zero finite complex number. If for each $f\in\mathcal{F}$, all zeros of $f$ have multiplicity at least $k+2,$ and $f^{(k)}(z)\neq b$ on $D,$ then $\mathcal{F}$ is normal on $D.$

Wang and Fang  [11] gives an example to show that Theorem Theorem A, is not valid if all zeros of $f$ have multiplicity less than $k+2.$

By the ideas of shared values, M. Fang and L. Zalcman  [2, 3] proved

Theorem B**.**

Suppose that $k$ is a positive integer and $b\neq 0$ be a finite complex number. Let $\mathcal{F}$ be a family of meromorphic function in a domain $D$, all of zeros of $f\in\mathcal{F}$ are of multiplicity at least $k+2.$ If for each $f,g\in\mathcal{F},f$ and $g$ share $0,f^{(k)}$ and $g^{(k)}$ share $b$ IM in $D$, then $\mathcal{F}$ is normal in $D.$

In 2009, Y. Li and Y. Gu  [6] proved the following result.

Theorem C**.**

Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain $D.$ Let $k,n\geq k+2$ be positive integers and $b\neq 0$ be a finite complex number. If for each pair of functions $f,g\in\mathcal{F},$ $(f^{n})^{(k)}$ and $(g^{n})^{(k)}$ share $b$ in $D$, then $\mathcal{F}$ is normal in $D.$

Recently, releasing the condition that poles of $f(z)$ are of multiplicity at least $k+2$, J. Qui and T. Zhu [8] proved the following.

Theorem D**.**

Let $k\geq 2$ be an integer and let $b$ be a non zero finite complex number. Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain $D,$ such that for each $f\in\mathcal{F}$, all zeros of $f(z)$ have multiplicity at least $k+2,$ and all zeros of $f^{(k)}(z)$ are multiple. If for each $f,g\in\mathcal{F}$, $f$ and $g$ share $b$ in $D,$ then $\mathcal{F}$ is normal in $D.$

It is natural to ask whether Theorem D. can be improved by the idea of sharing a holomorphic function. In this paper we study this problem and obtain the following result.

Theorem 1.1**.**

Let $d\geq 0$ and $k\geq 2d+2$ be two integers and let $h\not\equiv 0$ be a holomorphic function in $D,$ and multiplicity of its all zeros is at most $d.$ Let $\mathcal{F}$ be a family of meromorphic functions in a domain $D$. If for each $f\in\mathcal{F},$ the multiplicity of all zeros of $f$ is at least $k+2d+2,$ and multiplicity of all zeros of $f^{(k)}$ is at least $2d+2.$ If for each pair of functions $f,g\in\mathcal{F},f$ and $g$ share $h$ in $D,$ then $\mathcal{F}$ is normal in $D.$

2. Some Lemmas

In order to prove our results we need the following lemmas. The well known Zalcman Lemma is a very important tool in the study of normal families. The following is a new version due to Zalcman  [14] (also see [13], p. $814$).

Lemma 2.1**.**

Let $\mathcal{F}$ be a family of meromorphic functions in the unit disk $\Delta$, with the property that for every function $f\in\mathcal{F},$ the zeros of $f$ are of multiplicity at least $l$ and the poles of $f$ are of multiplicity at least $k$. If $\mathcal{F}$ is not normal at $z_{0}$ in $\Delta$, then for $-l<\alpha<k$, there exist

1. (1)

a sequence of complex numbers $z_{n}\rightarrow z_{0}$, $|z_{n}|<r<1$, 2. (2)

a sequence of functions $f_{n}\in\mathcal{F}$, 3. (3)

a sequence of positive numbers $\rho_{n}\rightarrow 0$,

such that $g_{n}(\zeta)=\rho_{n}^{\alpha}f_{n}(z_{n}+\rho_{n}\zeta)$ converges to a non-constant meromorphic function $g$ on $\mathbb{C}$ with $g^{\#}(\zeta)\leq g^{\#}(0)=1$. Moreover, $g$ is of order at most two. Here, $g^{\#}(z)=\frac{|g^{\prime}(z)|}{1+|g(z)|^{2}}$ is the spherical derivative of $g$.

Lemma 2.2**.**

[7]** Let $f(z)$ be a transcendental meromorphic function of finite order on $\mathbb{C},$ and let $p(z)\not\equiv 0$ be a polynomial. Suppose that all zeros of $f(z)$ have multiplicity at least $k+1$, then $f^{(k)}(z)-p(z)$ has infinitely many zeros.

Lemma 2.3**.**

Let $d\geq 0$ and $k\geq 2d+2$ be two integers and let $p(z)\not\equiv 0$ be a polynomial of degree at most $d.$ Let $f(z)$ is a non constant rational function and multiplicity of all zeros of $f(z)$ is at least $k+2d+2$, and multiplicity of all zeros of $f^{(k)}(z)$ is at least $2d+2$. Then $f^{(k)}(z)-p(z)$ has at least two distict zeros and $f^{(k)}(z)-p(z)\not\equiv 0.$

Proof.

Case 1. Suppose that $f^{(k)}(z)-p(z)$ has exactly one zero at $z_{0}$ with multiplicity $l.$

Case 1.1. Suppose that $f$ is a non constant polynomial.

If $f^{(k)}(z)-p(z)\equiv 0$, then $f(z)$ is a polynomial of degree at most $k+d,$ which contradicts with the fact that multiplicity of all zeros of $f(z)$ is at least $k+2d+2$, Hence $f^{(k)}(z)-p(z)\not\equiv 0.$ Let

[TABLE]

where $K$ is a non-zero constant, $l$ is a positive integer. Because all zeros of $f^{(k)}(z)$ are of multiplicity at least $2d+2,$ we obtain $l\geq 2d+2,$ then

[TABLE]

and $f^{(k+d+1)}(z)=Kl(l-d)(z-z_{0})^{l-d-1}$. This implies that $f^{(k+d-1)}$ has exactly one zero $z_{0}.$ So $f^{(k+d)}$ has only the same zero $z_{0}$ too. Hence $f^{(k+d)}(z_{0})=0$, which contradicts with $f^{(k+d)}(z_{0})-C\neq 0.$

Case 1.2. Suppose that $f$ is a non-polynomial rational function.

Since $g(z)$ is a rational function and not a polynomial, then obviously $f^{(k)}(z)-p(z)\not\equiv 0.$ Let

[TABLE]

where $A$ is a non zero constant. Since all zeros of $f^{(k)}(z)$ are of multiplicity at least $2d+2,$ we find $m_{i}\geq 2d+2(i=1,2,\ldots,s),n_{j}\geq k+1(j=1,2,\ldots,t).$

Let us define

[TABLE]

Differentiating both sides of (2.3) step by step, we obtain

[TABLE]

where $g_{1}(z)=(M-N)\ldots(M-N-d)z^{(s+t-1)(d+1)}+a_{t-1}z^{t-1}+\ldots+a_{0}(a_{t-1},\ldots a_{0}$ are constants).

Since $f^{(k)}(z)-p(z)$ has exactly one zero at $z_{0}.$ From (2.3), we get

[TABLE]

Now we consider the following cases:

Case 1.2.1. When $d\geq l$. Differentiating both sides of (2.6), $(d+1)$- times, we get

[TABLE]

where $g_{2}(z)=(l-N)\ldots(l-N-d)z^{(d+1)t-(d-l+1)}+\ldots b_{1}z+b_{0}$.

From (2.3) and(2.6), we get $d+N=M$. This implies $M\geq N$. Now from (2.5) and (2.7), we obtain

$M-(d+1)s\leq(d+1)t-(d-l+1)<(d+1)t$,

It follows that

$M<(d+1)(s+t)\leq(d+1)\left(\frac{M}{2d+2}+\frac{M}{2d+3}\right)<M.$

Which is a contradiction.

Case 1.2.2. When $d<l.$

Differentiating both sides of (2.6), $(d+1)$- times, we get

[TABLE]

where $g_{3}(z)=(l-N)\ldots(l-N-d)z^{(d+1)t}+\ldots c_{1}z+c_{0}$.

Differentiating (2.6) $d$-times, we get $z_{0}$ is a zero of $f^{(k+d)}(z)-p^{(d)}(z)$, as $p^{(d)}(z)\neq 0,$ then $z_{0}\neq\alpha_{i}(i=1,2,\ldots,s).$

Subcase 1.2.2.1 When $l<d+N$.

From (2.3) and (2.6), we get $M\geq N$. Since $z_{0}\neq\alpha_{i}$ for any $i\in\{1,2,\ldots,s\}$, therefore from (2.5) and (2.8), we get

$M-(d+1)s\leq(d+1)t,$

It follows that

$M\leq\frac{4d+5}{4d+6}M<M,$ Which is a contradiction.

Subcase 1.2.2.2 When $l\geq d+N.$

If $M>N,$ then similar to the proof of Subcase 1.2.2.1, we get a contradiction. Thus $M\leq N.$ Since $z_{0}\neq\alpha_{i}$ for any $i\in\{1,2,\ldots,s\}$, then from (2.5) and (2.8), we get

$l-d-1\leq(d+1)(s+t-1),$

It follows that

$N\leq(d+1)\left(\frac{N}{2d+2}+\frac{N}{2d+3}\right)<N$, which is a contradiction.

Case 2. If $f^{(k)}(z)-p(z)$ has no zero. Similar to case 1, we obtain $f^{(k)}(z)-p(z)\not\equiv 0.$

Now put $l=0$ in (2.1) and (2.6), and similar discussion to case 1, we get a contradiction.

Hence by case 1 and case 2, $f^{(k)}(z)-p(z)$ has at least two distinct zeros and $f^{(k)}(z)-p(z)\not\equiv 0.$ ∎

3. Proof of Main Result

Proof of Theorem 1.1.

Since normality is a local property, it is enough to show that $\mathcal{F}$ is normal at each $z_{0}\in D.$ we assume that ${D}=\Delta$. For each $z_{0}\in D$, either $h(z_{0})=0$ or $h(z_{0})\neq 0.$ Without loss of generality, we may assume that $z_{0}=0$.

Case 1. We first prove that $\mathcal{F}$ is normal at points $z$, where $h(z)=0$. By making standard normalization, we suppose that

[TABLE]

where $l\geq 1,\ b(0)=1,$ and $h(z)\neq 0$ when $0<|z|<1$. Let

[TABLE]

We shall prove that $\mathcal{F}_{1}$ is normal at $0.$

Suppose that $\mathcal{F}_{1}$ is not normal at $0,$ then by lemma 2.1, there exist $z_{j}\in\Delta$ tending to [math], functions $F_{j}\in\mathcal{F}_{1},$ positive numbers $\rho_{j}$ tending to [math], such that

[TABLE]

locally uniformly on $\mathbb{C}$ with respect to the sherical metric, where $g(\xi)$ is a non- constant meromorphic function on $\mathbb{C}$, whose order is at most 2. We distinguish two cases.

Case 1.1. There exist a subsequence of $\frac{z_{n}}{\rho_{n}},$ we still denote the subsequence by $\frac{z_{n}}{\rho_{n}}$, such that $\frac{z_{n}}{\rho_{n}}\rightarrow\alpha$, where $\alpha$ is a finite complex number. Then,

[TABLE]

spherically locally uniformly in $\mathbb{C}$. Then

[TABLE]

spherically locally uniformly in $\mathbb{C}$.

Since for all $f\in\mathcal{F}$ multiplicity of all zeros of $f$ is at least $k+2d+2$, and multiplicity of all zeros of $f^{(k)}$ is at least $2d+2,$ which implies multiplicity of all zeros of $G$ is at least $k+2d+2$, and by Hurwitz’s theorem all zeros of $G^{(k)}$ is at least $2d+2,$ then by lemma (2.1) and (2.2), $G^{(k)}(\xi)-\xi^{l}$ has at least two distinct zeros.

We claim that $G^{(k)}(\xi)-\xi^{l}$ has just a unique zero.

Suppose that $\xi_{0}$ and $\xi_{0}^{*}$ are two distinct zeros of $G^{(k)}(\xi)-\xi^{l}$, and choose $\delta(>0)$ small enough such that $D(\xi_{0},\delta)\cap D(\xi_{0}^{*},\delta)=\phi,$ where $D(\xi_{0},\delta)=\{\xi||\xi-\xi_{0}|<\delta\}$ and $D(\xi_{0}^{*},\delta)=\{\xi||\xi-\xi_{0}^{*}|<\delta\}$.

From (3.2), and by Hurwitz’s theorem, there exists points $\xi_{n}\in D(\xi_{0},\delta),\xi_{n}^{*}\in D(\xi_{0}^{*},\delta)$ such that for sufficiently large $n$

[TABLE]

[TABLE]

By the assumption that for each pair $f,g\in\mathcal{F},$ $f^{(k)}$ and $g^{(k)}$ share $h$ in $D$, we know that for any integer $m$

[TABLE]

[TABLE]

we fix $m$ and note that $\rho_{n}\xi_{n}\rightarrow 0,$ $\rho_{n}\xi_{n}^{*}\rightarrow 0,$ as $n\rightarrow\infty.$ From this we obtain

[TABLE]

Since the zeros of $f_{m}^{(k)}(z)-h(z)=0$ has no accumulation point, when $n$ is sufficiently large enough, we have

[TABLE]

Hence

[TABLE]

which contradicts with the fact that $\xi_{n}\in D(\xi_{0},\delta),\xi_{n}^{*}\in D(\xi_{0}^{*},\delta)$ and $D(\xi_{0},\delta)\cap D(\xi_{0}^{*},\delta)=\phi$.

Case 1.2. There exist a subsequence of $\frac{z_{n}}{\rho_{n}},$ we still denote the subsequence by $\frac{z_{n}}{\rho_{n}}$, such that $\frac{z_{n}}{\rho_{n}}\rightarrow\infty$. By simple calculation we obtain,

[TABLE]

where

[TABLE]

From (3) and the identity $\rho_{n}^{m}g_{n}^{(k-m)}=F_{n}^{(k-m)}(z_{n}+\rho_{n}\xi)$, we get

[TABLE]

Hence,

[TABLE]

Thus, we have

[TABLE]

spherically uniformly on compact subset of $\mathbb{C}$ disjoint from the poles of $g.$

Since for all $f\in\mathcal{F}$, all zeros of $f(z)$ have multiplicity at least $k+2d+2$, hence all zeros of $g(\xi)$ have multiplicity at least $k+2d+2$. Noting that all zeros of $f^{(k)}(z)$ have multiplicity at least $2d+2$. By Hurwitz’s theorem, all zeros of $g^{(k)}(\xi)$ have multiplicity at least $2d+2.$ Thus by lemma 2.1 and 2.2, $g^{(k)}(\xi)-1$ has at least two distinct zeros.

We claim that $g^{(k)}(\xi)-1$ has just a unique zero.

Suppose that $\xi_{1}$ and $\xi_{1}^{*}$ are two distinct zeros of $g^{(k)}(\xi)-1$, and choose $\delta(>0)$ small enough such that $D(\xi_{1},\delta)\cap D(\xi_{1}^{*},\delta)=\phi,$ where $D(\xi_{1},\delta)=\{\xi||\xi-\xi_{1}|<\delta\}$ and $D(\xi_{1}^{*},\delta)=\{\xi||\xi-\xi_{1}^{*}|<\delta\}$.

From (3.5), and by Hurwitz’s theorem, there exists points $\hat{\xi_{n}}\in D(\xi_{1},\delta),\hat{\xi_{n}^{*}}\in D(\xi_{1}^{*},\delta)$ such that for sufficiently large $n$

[TABLE]

[TABLE]

Similar to the proof of case 1, we get a contradiction. Hence $\mathcal{F}_{1}$ is normal at $0.$ It remains to prove that $\mathcal{F}$ is normal at 0.

Since $\mathcal{F}_{1}$ is normal at [math], then there exist $r>0$ and a subsequence $F_{n_{k}}$ of $F_{n}$ such that $F_{n_{k}}$ converges sherically uniformally to a meromorphic function $F(z)$ or $\infty.$ in $\Delta_{r}$

Now we consider two cases:

case i. When $f_{n_{k}}(0)\neq 0$, for $k$ large enough. Then $F(0)=0,$ then there exist $0<\delta<r$ such that $|F(z)|\geq 1$ in $\Delta(0,\delta)$. Thus $|F_{n_{k}}|>\frac{1}{2}$ in $\Delta(0,\delta)$, for sufficiently large $k.$ Hence $f_{n_{k}}$ is holomorphic in $\Delta(0,\delta.)$ Therefore ,

[TABLE]

By the Maximum principle and Montel’s theorem, $\mathcal{F}$ is normal at $0,$ and thus $\mathcal{F}$ is normal in $D.$

Case ii. When $f_{n_{k}}(0)=0$, for $k$ large enough. Since the multiplicity of all zeros of $f\in\mathcal{F}$ is at least $k+2d+2,$ then $F(0)=0.$ Hence, there exist $0<\rho<r$ such that $F(z)$ is holomorphic in $\Delta_{\rho}$. Hence $F_{n_{k}}$ converges spherically locally uniformly to a holomorphic function $F(z)$ in $\Delta_{\rho}$, hence, $f_{n_{k}}$ converges spherically locally uniformly to a holomorphic function $z^{l}F(z)$ in $\Delta_{\rho}.$ Hence $\mathcal{F}$ is normal at [math], and thus $\mathcal{F}$ is normal in $D.$

Case 2. Now we prove that $\mathcal{F}$ is normal at points $z$, where $h(z)\neq 0.$

Suppose that $\mathcal{F}$ is not normal at $z_{0},$ then by lemma 2.1, there exist $z_{j}\in\Delta$ tending to [math], functions $f_{n}\in\mathcal{F},$ positive numbers $\rho_{j}$ tending to $z_{0}$, such that

[TABLE]

locally uniformly on $\mathbb{C}$ with respect to the sherical metric, where $g(\xi)$ is a non- constant meromorphic function on $\mathbb{C}$, the multiplicity of all zeros of $g$ is at least $k+2d+2$, and multiplicity of all zeros of $g^{(k)}$ is at least $2d+2.$

Hence by lemma 2.2 and lemma 2.3, $g^{(k)}(\xi)-h(z_{0})$ has at least two distinct zeros, and $g^{(k)}(\xi)-h(z_{0})\not\equiv 0.$

Similar to the proof of case 1, we get a contradiction. Hence $\mathcal{F}$ is normal at $z_{0}.$

Since $z_{0}$ is arbitrary, thus $\mathcal{F}$ is normal in $D.$ ∎