1 Introduction and main results

A family \(\mathcal F\) of meromorphic functions defined on a domain D is said to be normal in D if every sequence of elements of \(\mathcal F\) contains a subsequence which converges locally uniformly on D with respect to the spherical metric to a meromorphic function or \(\infty \) (see [8]). One of the interesting quantities characterizing the normal families of meromorphic functions is the spherical derivative. The spherical derivative of a meromorphic function f(z) is defined to be

$$\begin{aligned} f^{\#}(z)=\frac{\left| f'(z)\right| }{1 + \left| f(z)\right| ^2 }, \end{aligned}$$

with an obvious modification if \(f(z)=\infty \). By well known result of Marty, normality of any family of meromorphic functions on some domain is equivalent to local boundedness of the corresponding family of spherical derivatives. The following significant improvement of one direction Marty’s theorem due to Hinkkanen [3] and Lappan [5] allows us to reduce drastically the set on which spherical derivatives are required to be bounded.

Theorem 1.1

A family \(\mathcal F\) of meromorphic functions on a domain \(D \subset \mathbb {C}\) is normal if and only if for each compact set \(K \subset D\), there exist a set \( S = S(K) \subset \overline{\mathbb {C}}\) containing at least five distinct points and a positive constant \(M = M(K)\) such that

$$\begin{aligned} \max _{z \in K \cap f^{- 1}(S)}f^{\#}(z) \le M,\;\;f \in \mathcal F. \end{aligned}$$

An analogous five-point theorem for normal function was earlier proved by Lappan [4]: Let S be any set consisting of five distinct complex numbers. If f is a meromorphic function on the unit disk \(\mathbb {D}\) such that

$$\begin{aligned} \sup \left\{ (1-\left| z\right| ^2)f^{\#}(z): z \in f^{-1} (S)\right\} < \infty , \end{aligned}$$

then f is a normal function.

Regarding the cardinality of set S in Theorem 1.1, Lappan [4] showed that the number “five” cannot be replaced by “three” and there are at least some cases in which “five” cannot be replaced by “four”.

Definition 1.2

Let f be a meromorphic function on a domain D and S be a set of n-distinct meromorphic functions on D. Then, for \(z \in D\) we write \([f \in S]_z\) if \(f(z)= \varphi (z)\) for some \(\varphi \in S\).

In this paper we extend Theorem 1.1 by replacing the elements of set S by distinct meromorphic functions and hence obtain a generalization of Lappan’s five point theorem.

Theorem 1.3

Let \(\mathcal {F}\) be a family of meromorphic functions on a domain \(D \subset \mathbb {C}\) and let \(S=\left\{ \varphi _i:1\le i \le 5\right\} \) be a set of five distinct meromorphic functions on D. If for every \(f \in \mathcal {F}\),

  1. 1.

    there is a constant \(M > 0\) such that

    $$\begin{aligned}{}[f \in S]_z \Rightarrow f^{\#}(z) \le M,\;\text {and} \end{aligned}$$
  2. 2.

    \(f(z_0)\ne \varphi (z_0)\) for all \(\varphi \in S\) whenever \(\varphi _i(z_0)=\varphi _j(z_0)\) for \(i,j \in \left\{ 1,2,3,4,5\right\} (i\ne j)\) and \(z_0 \in D\),

then \(\mathcal {F}\) is normal on D.

Example 1.4

Consider the family \(\mathcal F = \left\{ f_j: j \in \mathbb {N}\right\} \) and \(S = \left\{ \varphi _l: 1\le l \le 5 \right\} \), where

$$\begin{aligned} f_j(z) = 2j z^2 ~~~ \text{ and } ~~~ \varphi _l(z)= \frac{z^2}{l} \end{aligned}$$

on the open unit disk \(\mathbb {D}\). Clearly, for every \(f \in \mathcal F\),

$$\begin{aligned}{}[f \in S]_z \Rightarrow f^{\#}(z) \le M, \end{aligned}$$

for some constant \(M>0\). However, the family \(\mathcal F\) is not normal on \(\mathbb {D}\). Note that \(f_j(0)=\varphi _l(0)\), for \(1\le l \le 5\), showing that we cannot drop the condition (2) in Theorem 1.3. Moreover, replacing any one of the function in set S by constant \(\infty \) will also work in this example.

Taking somewhat greater effort in reducing the cardinality of set S in Theorem 1.1, Tan and Thin [9] obtained the following two results for the case where the spherical derivatives of \(f,\,f',\,f''\) are bounded above.

Theorem 1.5

Let \(\mathcal F\) be a family of meromorphic functions on a domain \(D \subset \mathbb {C}\). Assume that for each compact set \(K \subset D\), there exist a set \( S = S(K) \subset \overline{\mathbb {C}}\) containing four distinct points and a positive constant \(M = M(K)\) such that

$$\begin{aligned} \max _{z \in K \cap f^{- 1}(S)}f^{\#}(z) \le M\;\text { and }\;\max _{z \in K \cap f^{- 1}(S\setminus \{\infty \})}(f')^{\#}(z) \le M, \end{aligned}$$

for all \(f \in \mathcal F\). Then \(\mathcal F\) is normal.

Theorem 1.6

Let \(\mathcal F\) be a family of meromorphic functions on a domain \(D \subset \mathbb {C}\). Assume that for each compact set \(K \subset D\), there exist a set \( S = S(K) \subset \mathbb {C}\) containing three distinct points and a positive constant \(M = M(K)\) such that

$$\begin{aligned} f^{\#}(z) \le M,~~ (f')^{\#}(z) \le M \;\;\text {and}\;\;(f'')^{\#}(z) \le M, \end{aligned}$$

for all \(f \in \mathcal F\) and \(z \in K \cap f^{- 1}(S)\). Then \(\mathcal F\) is normal.

Motivated by the results of Tan and Thin, it is natural to ask whether one can reduce the cardinality of set S in Theorem 1.3 under some conditions. We investigate this situation and hence able to prove the following more general version of Theorem 1.3.

Theorem 1.7

Let m and n be integers with \(m \ge 1\), \(n \ge 3\) and \(m+n \le 6 \). Let \(\mathcal {F}\) be a family of meromorphic functions on a domain \(D \subset \mathbb {C}\), and let \(S = \{\varphi _1,\varphi _2,\cdots ,\varphi _n\}\) be a set of n-distinct meromorphic functions on D. Suppose that

$$\begin{aligned} E_f = \{z \in D:f(z) = \varphi _i(z) \ne \infty \}\;\text {and}\;F_f = \{z \in D:f(z) = \varphi _i(z) = \infty \}, \end{aligned}$$

for every \(f \in \mathcal F\) and \(i = 1,2, \cdots , n\). If for every \(f \in \mathcal {F}\),

  1. 1.

    there is a constant \(M > 0\) such that

    $$\begin{aligned} (f^{(k)})^{\#}(z) \le M\,\text { for }\,z \in E_f,~~~~~~~\left( \left( \frac{1}{f}\right) ^{(k)}\right) ^{\#}(z) \le M\,\text { for }\,z \in F_f, \end{aligned}$$

    where \(0 \le k \le m - 1\), and

  2. 2.

    for every point \(z_0 \in D\) the cardinality of the set \(\left\{ \varphi _1(z_0),\varphi _2(z_0),\cdots ,\varphi _n(z_0)\right\} \) is at most 2 implies that \(f(z_0)\ne \varphi _i(z_0)\) for at least 2 functions \(\varphi _i\),

then \(\mathcal {F}\) is normal on D.

It is worthwhile to mention the special case of Theorem 1.7: If \(n=5\) and \(m=1\), then by using the fact \(f^{\#}=\left( \frac{1}{f}\right) ^{\#}\), we see that Theorem 1.7 reduces to Theorem 1.3.

Example 1.8

Consider the family \(\mathcal F = \left\{ f_j: j \in \mathbb {N}\right\} \) and \(S = \left\{ 0, \infty \right\} \), where \(f_j(z) = e^{jz} \) on the open unit disk \(\mathbb {D}\). Clearly, the conditions (1) and (2) in Theorem 1.7 are satisfied. However, the family \(\mathcal F\) is not normal on \(\mathbb {D}\). This shows that the cardinality of set S in Theorem 1.7 cannot be reduced.

Example 1.9

Consider the family \(\mathcal F = \left\{ f_j: j \in \mathbb {N}\right\} \) and \(S = \left\{ -i,0, i, \right\} \), where

$$f_j(z) = tan~jz $$

on the open unit disk \(\mathbb {D}\). Clearly, \([f \in S]_0\) for every \(f \in \mathcal F\) and \((f_j)^{\#}(0) \rightarrow \infty \) as \(j \rightarrow \infty \). However, the family \(\mathcal F\) is not normal on \(\mathbb {D}\). This shows that the condition (1) is essential in Theorem 1.7.

Finally from Theorem 1.7 we obtain the following corollary by setting \(F_f = \phi \) for every \(f \in \mathcal F\).

Corollary 1.10

Let m and n be integers with \(m \ge 1\), \(n \ge 3\) and \(m+n \le 6 \). Let \(\mathcal {F}\) be a family of meromorphic functions on a domain \(D \subset \mathbb {C}\) and let \(S = \{\varphi _1,\varphi _2,\cdots ,\varphi _n\}\) be a set of n-distinct holomorphic functions on D. If for every \(f \in \mathcal {F}\),

  1. 1.

    there is a constant \(M > 0\) such that

    $$\begin{aligned}{}[f \in S]_z \Rightarrow (f^{(k)})^{\#}(z) \le M,\; 0 \le k \le m - 1,\;\text {and} \end{aligned}$$
  2. 2.

    for every point \(z_0 \in D\) the cardinality of the set \(\left\{ \varphi _1(z_0),\varphi _2(z_0),\cdots ,\varphi _n(z_0)\right\} \) is at most 2 implies that \(f(z_0)\ne \varphi _i(z_0)\) for at least 2 functions \(\varphi _i\),

then \(\mathcal {F}\) is normal on D.

One can easily see that Theorem 1.6 is a special case of Corollary 1.10 when the set S contains only three distinct points.

2 Proof of the main result

In order to prove our main result we need the famous rescaling lemma which was originally proved by Zalcman [10] and later extended by Pang [6, 7], and by Chen and Gu [2]. Here we present the following general version of this rescaling lemma:

Lemma 2.1

(Zalcman–Pang Lemma) Let \(\mathcal F\) be a family of meromorphic functions on \(\mathbb {D}\) all of whose zeros and poles have multiplicity at least l, p respectively. Then \(\mathcal F\) is not normal at a point \(z_0 \in \mathbb {D}\) if and only if there exist, for each \(\alpha :-p<\alpha <l\),

  1. (i)

    a real number r: \(0<r <1\),

  2. (ii)

    points \(z_n\): \(\left| z_n\right| <r\),

  3. (iii)

    positive numbers \(\rho _n\): \(\rho _n\rightarrow \)0,

  4. (iv)

    functions \(f_n\in \mathcal F\)

such that \(g_n(\zeta )=\rho _n ^{-\alpha } f_n(z_n+\rho _n \zeta )\) converges locally uniformly with respect to the spherical metric to g(\(\zeta \)), where \(g(\zeta \)) is a non-constant meromorphic function on \(\mathbb {C}\) all of whose zeros and poles have multiplicity at least l, p respectively. Moreover, \(g^{\#}\)(\(\zeta \))\(\le g^{\#}(0)=1\) and g has order at most 2.

Furthermore, we require the following normality criterion due to Chang et al. [1].

Lemma 2.2

[1] Let \(\mathcal F\) be a family of meromorphic functions on a domain \(D \subset \mathbb {C}\) and let a and b be distinct functions holomorphic on D. Suppose that, for any \(f\in \mathcal {F}\) and any \(z\in D\), \(f(z)\ne a(z)\) and \(f(z)\ne b(z)\). If \(\mathcal F\) is normal on \(D-\left\{ 0\right\} \), then \(\mathcal F\) is normal on D.

Proof of Theorem 1.7

Since normality is a local property, it is enough to show that \(\mathcal F\) is normal at each \(z_0 \in D\). Let \(S_1 = \{\varphi _1(z_0), \varphi _2(z_0), \cdots ,\varphi _n(z_0)\}\). Without loss of generality we can assume that all the values in \(S_1\) are finite, otherwise we can choose a finite value \(c\notin \left\{ \varphi _1(z_0),\varphi _2(z_0),\cdots ,\varphi _n(z_0)\right\} \) and turn to prove the normality of the family \(\left\{ 1/(f-c), f \in \mathcal F\right\} \). Now, we distinguish the following cases:

Case 1. When cardinality of \(S_1\) is at least three.

Suppose that \(\mathcal {F}\) is not normal at \(z_0\). Then by Lemma 2.1, for \(\alpha =0 \) we can find a sequence \(\left\{ f_j \right\} \) in \(\mathcal {F}\), a sequence \(\left\{ z_j\right\} \) of complex numbers with \(z_j\rightarrow z_0\) and a sequence \(\left\{ \rho _j\right\} \) of positive real numbers with \(\rho _j \rightarrow 0\) such that

$$\begin{aligned} g_j(\zeta ) = f_j(z_j + \rho _j \zeta ) \end{aligned}$$

converges locally uniformly with respect to the spherical metric to a non-constant meromorphic function \(g(\zeta )\) on \(\mathbb {C}\) such that \({{g}^{\#}}(\zeta )\le {{g}^{\#}}(0)=1\) for all \(\zeta \in \mathbb {C}\). Therefore, for every \(k \in \mathbb {N}\), we have \(g^{(k)}_j \longrightarrow g^{(k)} ~~\text {on~~} \mathbb {C}-P\) locally uniformly with respect to the Euclidean metric, where P is the pole set of g. Since g is a non-constant meromorphic function on \(\mathbb {C}\), by Picard’s theorem g assumes at least one of the values of \(S_1\). Let \(\zeta _0 \in \mathbb {C}\) be such that \(g(\zeta _0) - \varphi _i(z_0) = 0\), for some \(i=1,2,\cdots , n \). Since \(g(\zeta ) \not \equiv \varphi _i(z_0)\), by Hurwitz’s theorem there exist a sequence of points \(\left\{ \zeta _j\right\} \rightarrow \zeta _0\) such that for sufficiently large j,

$$\begin{aligned} g_j(\zeta _j) = f_j(z_j + \rho _j \zeta _j) = \varphi _i(z_j + \rho _j \zeta _j). \end{aligned}$$

By hypothesis, for every \(f \in \mathcal {F}\), \([f \in S]_z \Rightarrow (f^{(k)})^{\#}(z) \le M\) (\(k = 0, 1, \cdots , m-1\)), it follows that

$$(f_j^{(k)})^{\#}(z_j + \rho _j\zeta _j) \le M$$
$$\begin{aligned} \Rightarrow \frac{|f_j^{(k + 1)}(z_j + \rho _j\zeta _j)|}{1 + |f_j^{(k)}(z_j + \rho _j\zeta _j)|^2} \le M, \end{aligned}$$
(2.1)

for all \(k = 0,1,\cdots ,m - 1\) and for all j sufficiently large.

Set

$$\begin{aligned} M_1:= M \cdot (1+ (1+\left| \varphi _i(z_0)\right| ^2)) ~~\text {and}~~ M_{k+1}:=M \cdot (1+{M_k}^2). \end{aligned}$$

We claim that

$$\begin{aligned} |f_j^{(k)}(z_j + \rho _j\zeta _j)| \le M_k,\;\text {for}\;k= 1, 2, \cdots , m. \end{aligned}$$
(2.2)

We shall prove this claim by using the method of induction.

From (2.1), we have

$$\begin{aligned}{} & {} \frac{|f_j'(z_j + \rho _j\zeta _j)|}{1 + |f_j(z_j + \rho _j\zeta _j)|^2} \le M\nonumber \\{} & {} \Rightarrow |f_j'(z_j + \rho _j\zeta _j)| \le M(1 + |f_j(z_j + \rho _j\zeta _j)|^2). \end{aligned}$$
(2.3)

Since \(\varphi _i(z_j + \rho _j\zeta _j) \rightarrow \varphi _i(z_o) \ne \infty \), we may assume that

$$\begin{aligned} |f_j(z_j + \rho _j\zeta _j)| = |\varphi _i(z_j + \rho _j\zeta _j)| \le 1 + |\varphi _i(z_o)|. \end{aligned}$$

By using this in (2.3), we have

$$\begin{aligned} |f_j'(z_j + \rho _j\zeta _j)| \le M(1 + (1 + |\varphi _i(z_o)|^2)) = M_1. \end{aligned}$$

This proves our claim for \(k = 1\). Assume that (2.2) holds for some \(k\,(k \le m - 1)\). Then by (2.1) and by induction hypothesis, we have

$$\begin{aligned} |f_j^{(k + 1)}(z_j + \rho _j\zeta _j)|&\le M(1 + |f_j^{(k)}(z_j + \rho _j\zeta _j)|^2)\\&\le M\,(1 + M_k^2) = M_{k + 1}. \end{aligned}$$

Hence, by induction, we get (2.2).

Now, by (2.2), we have

$$\begin{aligned} \frac{|g_j^{(k)}(\zeta _j)|}{1 + |g_j^{(k - 1)}(\zeta _j)|^2}&= \rho _j^k\,\frac{|f_j^{(k)}(z_j + \rho _j\zeta _j)|}{1 + \rho _j^{2(k - 1)}|f_j^{(k - 1)}(z_j + \rho _j\zeta _j)|}\\&\le \rho _j^k |f_j^{(k)}(z_j + \rho _j\zeta _j)|\\&\le \rho _j^k\,M_k, \end{aligned}$$

for all \(k = 1,2, \cdots , m.\)

Since \(g_j^{(k - 1)}(\zeta _j) \rightarrow g^{(k - 1)}(\zeta _o) \ne \infty \), from above inequality, we get

$$\begin{aligned} 0 \le |g^{(k)}(\zeta _o)| = \underset{j\rightarrow \infty }{\mathop {\lim }}|g^{(k)}_j(\zeta _j)| \le \underset{j\rightarrow \infty }{\mathop {\lim }}\rho _j^k\,M_k(1 + |g^{(k - 1)}_j(\zeta _j)|^2) = 0. \end{aligned}$$

Thus \(g^{(k)}(\zeta _0) = 0\) for all \(k= 1,2,\cdots ,m\), and hence \(\zeta _0\) is a zero of \(g(\zeta ) - \varphi _i(z_0)\) of multiplicity at least \(m + 1\).

Now, by applying second fundamental theorem of Nevanlinna for \(n(\ge 3)\)-distinct values in \(S_1\), we have

$$\begin{aligned} (n - 2)T(r, g)&\le \overline{N}\left( r, \frac{1}{g - \varphi _1(z_0)}\right) + \overline{N}\left( r, \frac{1}{g - \varphi _2(z_0)}\right) + \cdots + \overline{N}\left( r, \frac{1}{g - \varphi _n(z_0)}\right) \nonumber \\&\quad + S(r, g).\nonumber \\&\le \frac{1}{m + 1} \left[ N\left( r, \frac{1}{g - \varphi _1(z_0)}\right) + N\left( r, \frac{1}{g - \varphi _2(z_0)}\right) + \cdots + N\left( r, \frac{1}{g - \varphi _n(z_0)}\right) \right] \nonumber \\&\quad + S(r, g)\nonumber \\&\le \frac{n}{m + 1}T(r, g) + S(r, g).\nonumber \end{aligned}$$

That is

$$\begin{aligned} \left( (n-2)- \left( \frac{n}{m+1} \right) \right) T(r,g) \le S(r,g). \end{aligned}$$

This is a contradiction to the fact that g is a non-constant meromorphic function. Hence \(\mathcal F\) is normal at \(z_0\).

Case 2. When cardinality of \(S_1\) is at most two.

By condition (2), there exist at least two functions \(\varphi _i\) for which \(f(z_0) \ne \varphi _i(z_0)\) for every \(f \in \mathcal F\). Moreover, we can find a small disk \(D_r(z_0)\) around \(z_0\) such that each \(\varphi _i\) is holomorphic with \(\varphi _i(z) \ne \varphi _j(z)\) \((1 \le i, j \le n)\) in \(D_r(z_0)-\left\{ z_0\right\} \). Thus by Case 1, \(\mathcal F\) is normal in \(D_r(z_0)-\left\{ z_0\right\} \).

Next we show that \(\mathcal F\) is normal at \(z_0\). Since for every \(f \in \mathcal F\), \(f(z_0) \ne \varphi _i(z_0)\) for at least two functions \(\varphi _i\) and each \(\varphi _i(z_0)\) is finite, we find that for every \(f \in \mathcal F\), \(f(z) \ne \varphi _i(z)\) for at least two functions \(\varphi _i\) which are holomorphic in \(D_r(z_0)\). Thus by Lemma 2.2, \(\mathcal F\) is normal at \(z_0\).

This completes the proof of Theorem 1.7.