A generalization of Lappan’s five point theorem

In this paper, we prove the following result: Let F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} be a family of meromorphic functions on a domain D and let S=φi:1≤i≤5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=\left\{ \varphi _i:1\le i \le 5\right\} $$\end{document} be a set of five distinct meromorphic functions on D. If for each f∈F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in \mathcal {F}$$\end{document} and z0∈D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0 \in D$$\end{document}, there is a constant M>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M>0$$\end{document} such that f#(z0)≤M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{\#}(z_0) \le M$$\end{document} whenever f(z0)=φ(z0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z_0)= \varphi (z_0)$$\end{document} for some φ∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in S$$\end{document} and if f(z0)≠φ(z0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z_0) \ne \varphi (z_0)$$\end{document} for all φ∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in S$$\end{document} whenever φi(z0)=φj(z0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _i(z_0) = \varphi _j(z_0) $$\end{document} for some i,j∈1,2,3,4,5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i,j \in \left\{ 1,2,3,4,5\right\} $$\end{document} with i≠j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i \ne j$$\end{document}, then F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} is normal on D. Further we extend this result to the case where the set S contains fewer functions. In particular, our result generalizes the most significant theorem of Lappan (i.e. Lappan’s five point theorem).


Introduction and main results
A family F of meromorphic functions defined on a domain D is said to be normal in D if every sequence of elements of F contains a subsequence which converges locally uniformly on D with respect to the spherical metric to a meromorphic function or ∞ (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 with an obvious modification if f (z) = ∞.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.
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 D such that 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 ∈ D we write 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 F be a family of meromorphic functions on a domain D ⊂ C and let S = {ϕ i : 1 ≤ i ≤ 5} be a set of five distinct meromorphic functions on D. If for every f ∈ F, 1. there is a constant M > 0 such that Example 1.4 Consider the family F = f j : j ∈ N and S = {ϕ l : 1 ≤ l ≤ 5}, where on the open unit disk D. Clearly, for every f ∈ F, for some constant M > 0. However, the family F is not normal on D. Note that f j (0) = ϕ l (0), for 1 ≤ l ≤ 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 ∞ 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 F be a family of meromorphic functions on a domain D ⊂ C. Assume that for each compact set K ⊂ D, there exist a set S = S(K ) ⊂ C containing four distinct points and a positive constant M
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 ≥ 1, n ≥ 3 and m + n ≤ 6.Let F be a family of meromorphic functions on a domain D ⊂ C, and let S = {ϕ 1 , ϕ 2 , • • • , ϕ n } be a set of n-distinct meromorphic functions on D. Suppose that where 0 ≤ k ≤ m − 1, and 2. for every point z 0 ∈ D the cardinality of the set {ϕ It is worthwhile to mention the special case of Theorem 1.7:If n = 5 and m = 1, then by using the fact , we see that Theorem 1.7 reduces to Theorem 1.3.
Example 1.8 Consider the family F = f j : j ∈ N and S = {0, ∞}, where f j (z) = e jz on the open unit disk D. Clearly, the conditions ( 1) and ( 2) in Theorem 1.7 are satisfied.However, the family F is not normal on D. This shows that the cardinality of set S in Theorem 1.7 cannot be reduced.
Example 1.9 Consider the family F = f j : j ∈ N and S = {−i, 0, i, }, where However, the family F is not normal on 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 = φ for every f ∈ F. 1. there is a constant M > 0 such that 2. for every point z 0 ∈ D the cardinality of the set {ϕ 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.

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 F be a family of meromorphic functions on D all of whose zeros and poles have multiplicity at least l, p respectively.Then F is not normal at a point z 0 ∈ D if and only if there exist, for each α : −p < α < l, (i) a real number r : converges locally uniformly with respect to the spherical metric to g(ζ ), where g(ζ ) is a non-constant meromorphic function on C all of whose zeros and poles have multiplicity at least l, p respectively.Moreover, g # (ζ )≤ 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 F be a family of meromorphic functions on a domain D ⊂ C and let a and b be distinct functions holomorphic on D. Suppose that, for any f ∈ F and any z ∈ D, f (z) = a(z) and f (z Without loss of generality we can assume that all the values in S 1 are finite, otherwise we can choose a finite value c / ∈ {ϕ 1 (z 0 ), ϕ 2 (z 0 ), • • • , ϕ n (z 0 )} and turn to prove the normality of the family {1/( f − c), f ∈ F}.Now, we distinguish the following cases: Case 1.When cardinality of S 1 is at least three.Suppose that F is not normal at z 0 .Then by Lemma 2.1, for α = 0 we can find a sequence f j in F, a sequence z j of complex numbers with z j → z 0 and a sequence ρ j of positive real numbers with ρ j → 0 such that converges locally uniformly with respect to the spherical metric to a non-constant meromorphic function g (k) on 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 C, by Picard's theorem g assumes at least one of the values of , by Hurwitz's theorem there exist a sequence of points ζ j → ζ 0 such that for sufficiently large j,

By hypothesis, for every
for all k = 0, 1, • • • , m − 1 and for all j sufficiently large.Set We shall prove this claim by using the method of induction.From (2.1), we have By using this in (2.3), we have This proves our claim for k = 1.Assume that (2.2) holds for some k (k ≤ m − 1).Then by (2.1) and by induction hypothesis, we have Hence, by induction, we get (2.2).Now, by (2.2), we have Now, by applying second fundamental theorem of Nevanlinna for n(≥ 3)-distinct values in S 1 , we have T (r, g) + S(r, g).
That is This is a contradiction to the fact that g is a non-constant meromorphic function.Hence 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 ϕ i for which f (z 0 ) = ϕ i (z 0 ) for every f ∈ F.Moreover, we can find a small disk D r (z 0 ) around z 0 such that each ϕ i is holomorphic with ϕ i (z) = ϕ j (z) (1 ≤ i, j ≤ n) in D r (z 0 ) − {z 0 }.Thus by Case 1, F is normal in D r (z 0 ) − {z 0 }.
Next we show that F is normal at z 0 .Since for every f ∈ F, f (z 0 ) = ϕ i (z 0 ) for at least two functions ϕ i and each ϕ i (z 0 ) is finite, we find that for every f ∈ F, f (z) = ϕ i (z) for at least two functions ϕ i which are holomorphic in D r (z 0 ).Thus by Lemma 2.2, F is normal at z 0 .
This completes the proof of Theorem 1.7. 123

Theorem 1 . 1
A family F of meromorphic functions on a domain D ⊂ C is normal if and only if for each compact set K ⊂ D, there exist a set S = S(K ) ⊂ C containing at least five distinct points and a positive constant M = M(K ) such that

Theorem 1 . 6
Let F be a family of meromorphic functions on a domain D ⊂ C. Assume that for each compact set K ⊂ D, there exist a set S = S(K ) ⊂ C containing three distinct points and a positive constant M

Corollary 1 . 10
Let m and n be integers with m ≥ 1, n ≥ 3 and m + n ≤ 6.Let F be a family of meromorphic functions on a domain D ⊂ C and let S = {ϕ 1 , ϕ 2 , • • • , ϕ n } be a set of n-distinct holomorphic functions on D. If for every f ∈ F,