Normality through sharing of pairs of functions with derivatives
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909380822171648 |
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| author | Charak, Kuldeep Singh Kumar, Manish Singh, Anil |
| author_facet | Charak, Kuldeep Singh Kumar, Manish Singh, Anil |
| contents | Let $\mathcal{F}\subset\mathcal{M}(D)$ and let $a, b$ and $c$ be three distinct complex numbers. If, there exist a holomorphic function $h$ on $D$ and a positive constant $ρ$ such that for each $f\in\mathcal{F},$ $f$ and $f^{'}$ partially share three pairs of functions $(a,h), \ (b, c_f)$ and $(c,d_f)$ on $D,$ where $c_f$ and $d_f$ are some values in some punctured disk $D^*_ρ(0),$ then $\mathcal{F}$ is normal in $D$. This is an improvement of Schwick's result[Arch. Math. (Basel), \textbf{59} (1992), 50-54]. We also obtain several normality criteria which significantly improve the existing results and examples are given to establish the sharpness of results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_04314 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Normality through sharing of pairs of functions with derivatives Charak, Kuldeep Singh Kumar, Manish Singh, Anil Complex Variables 30D30, 30D45 Let $\mathcal{F}\subset\mathcal{M}(D)$ and let $a, b$ and $c$ be three distinct complex numbers. If, there exist a holomorphic function $h$ on $D$ and a positive constant $ρ$ such that for each $f\in\mathcal{F},$ $f$ and $f^{'}$ partially share three pairs of functions $(a,h), \ (b, c_f)$ and $(c,d_f)$ on $D,$ where $c_f$ and $d_f$ are some values in some punctured disk $D^*_ρ(0),$ then $\mathcal{F}$ is normal in $D$. This is an improvement of Schwick's result[Arch. Math. (Basel), \textbf{59} (1992), 50-54]. We also obtain several normality criteria which significantly improve the existing results and examples are given to establish the sharpness of results. |
| title | Normality through sharing of pairs of functions with derivatives |
| topic | Complex Variables 30D30, 30D45 |
| url | https://arxiv.org/abs/2209.04314 |