A generalization of Lappan’s five point theorem
| Autor: | Virender Singh, Banarsi Lal |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Arabian Journal of Mathematics, Vol 12, Iss 3, Pp 697-702 (2023) |
| Druh dokumentu: | article |
| ISSN: | 2193-5343 2193-5351 |
| DOI: | 10.1007/s40065-023-00418-z |
| Popis: | Abstract In this paper, we prove the following result: Let $$\mathcal {F}$$ F be a family of meromorphic functions on a domain D and let $$S=\left\{ \varphi _i:1\le i \le 5\right\} $$ S = φ i : 1 ≤ i ≤ 5 be a set of five distinct meromorphic functions on D. If for each $$f \in \mathcal {F}$$ f ∈ F and $$z_0 \in D$$ z 0 ∈ D , there is a constant $$M>0$$ M > 0 such that $$f^{\#}(z_0) \le M$$ f # ( z 0 ) ≤ M whenever $$f(z_0)= \varphi (z_0)$$ f ( z 0 ) = φ ( z 0 ) for some $$\varphi \in S$$ φ ∈ S and if $$f(z_0) \ne \varphi (z_0)$$ f ( z 0 ) ≠ φ ( z 0 ) for all $$\varphi \in S$$ φ ∈ S whenever $$\varphi _i(z_0) = \varphi _j(z_0) $$ φ i ( z 0 ) = φ j ( z 0 ) for some $$i,j \in \left\{ 1,2,3,4,5\right\} $$ i , j ∈ 1 , 2 , 3 , 4 , 5 with $$i \ne j$$ i ≠ j , then $$\mathcal {F}$$ 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). |
| Databáze: | Directory of Open Access Journals |
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