On existence questions for the functional equations $f^8+g^8+h^8=1$ and $f^6+g^6+h^6=1$
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| Format: | Preprint |
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2024
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| _version_ | 1866912804905156608 |
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| author | Li, Xiao-Min Yi, Hong-Xun Korhonen, Risto |
| author_facet | Li, Xiao-Min Yi, Hong-Xun Korhonen, Risto |
| contents | In 1985, W.K.Hayman (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, 1984(1985), 1-13.) proved that there do not exist non-constant meromorphic functions $f,$ $g$ and $h$ satisfying the functional equation $f^n+g^n+h^n=1$ for $n\geq 9.$ We prove that there do not exist non-constant meromorphic solutions $f,$ $g,$ $h$ satisfying the functional equation $f^8+g^8+h^8=1.$ In 1971, N. Toda (Tôhoku Math. J. 23(1971), no. 2, 289-299.) proved that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying $f^n+g^n+h^n=1$ for $n\geq 7.$ We prove that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying the functional equation $f^6+g^6+h^6=1.$ Our results answer questions of G. G. Gundersen. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_01509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On existence questions for the functional equations $f^8+g^8+h^8=1$ and $f^6+g^6+h^6=1$ Li, Xiao-Min Yi, Hong-Xun Korhonen, Risto Complex Variables 30D30, 30D35 In 1985, W.K.Hayman (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, 1984(1985), 1-13.) proved that there do not exist non-constant meromorphic functions $f,$ $g$ and $h$ satisfying the functional equation $f^n+g^n+h^n=1$ for $n\geq 9.$ We prove that there do not exist non-constant meromorphic solutions $f,$ $g,$ $h$ satisfying the functional equation $f^8+g^8+h^8=1.$ In 1971, N. Toda (Tôhoku Math. J. 23(1971), no. 2, 289-299.) proved that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying $f^n+g^n+h^n=1$ for $n\geq 7.$ We prove that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying the functional equation $f^6+g^6+h^6=1.$ Our results answer questions of G. G. Gundersen. |
| title | On existence questions for the functional equations $f^8+g^8+h^8=1$ and $f^6+g^6+h^6=1$ |
| topic | Complex Variables 30D30, 30D35 |
| url | https://arxiv.org/abs/2411.01509 |