A generalization of Lappan’s five point theorem
Tóm tắt
In this paper, we prove the following result: Let
$$\mathcal {F}$$
be a family of meromorphic functions on a domain D and let
$$S=\left\{ \varphi _i:1\le i \le 5\right\} $$
be a set of five distinct meromorphic functions on D. If for each
$$f \in \mathcal {F}$$
and
$$z_0 \in D$$
, there is a constant
$$M>0$$
such that
$$f^{\#}(z_0) \le M$$
whenever
$$f(z_0)= \varphi (z_0)$$
for some
$$\varphi \in S$$
and if
$$f(z_0) \ne \varphi (z_0)$$
for all
$$\varphi \in S$$
whenever
$$\varphi _i(z_0) = \varphi _j(z_0) $$
for some
$$i,j \in \left\{ 1,2,3,4,5\right\} $$
with
$$i \ne j$$
, then
$$\mathcal {F}$$
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).
Tài liệu tham khảo
Chang, J.M.; Fang, M.L.; Zalcman, L.: Composite meromorphic functions and normal families. Proc. Roy. Soc. Edinb. Ser. A 139, 57–72 (2009)
Chen, H.H.; Lappan, P.: Spherical derivative, higher derivatives and normal families. Adv. Math. (China) 25(6), 517–524 (1996)
Hinkkanen, A.: Normal family and Ahlfor’s five islands theorem. N. Z. J. Math. 22, 39–41 (1993)
Lappan, P.: A criterion for a meromorphic function to be normal. Comment. Math. Helv. 49, 492–495 (1974)
Lappan, P.: A uniform approach to normal families. Rev. Roumaine Math. Pure Appl. 39, 691–702 (1994)
Pang, X.C.: Bloch principle and normality criterion. Sci. China 32(7), 782–791 (1989)
Pang, X.C.: On normal criterion of meromorphic functions. Sci. China 33(5), 521–527 (1990)
Schiff, J.L.: Normal Families. Springer-Verlag, New York (1993)
Tan, T.V.; Thin, N.V.: On Lappan, s five-point theorem. Comput. Methods Funct. Theory 17(1), 47–63 (2017)
Zalcman, L.: A heuristic principle in complex function theory. Am. Math. Month. 82, 813–817 (1975)