Some new findings concerning value distribution of a pair of delay-differential polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913863077724160 |
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| author | Long, Jianren Xiang, Xuxu |
| author_facet | Long, Jianren Xiang, Xuxu |
| contents | The paired Hayman's conjecture of different types are considered. More accurately speaking, the zeros of a pair of $f^nL(z,g)-a_1(z)$ and $g^mL(z,f)-a_2(z)$ are characterized using different methods from those previously employed, where $f$ and $g$ are both transcendental entire functions, $L(z,f)$ and $L(z,g)$ are non-zero linear delay-differential polynomials, $\min\{n,m\}\ge 2$, $a_1,a_2$ are non-zero small functions with relative to $f$ and $g$, or to $f^n(z)L(z,g)$ and $g^m(z)L(z,f)$, respectively. These results give answers to three open questions raised by Gao, Liu[Bull. Korean Math. Soc. 59 (2022)] and Liu, Liu[J. Math. Anal. Appl. 543 (2025)]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06825 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some new findings concerning value distribution of a pair of delay-differential polynomials Long, Jianren Xiang, Xuxu Complex Variables 39A05, 30D35 The paired Hayman's conjecture of different types are considered. More accurately speaking, the zeros of a pair of $f^nL(z,g)-a_1(z)$ and $g^mL(z,f)-a_2(z)$ are characterized using different methods from those previously employed, where $f$ and $g$ are both transcendental entire functions, $L(z,f)$ and $L(z,g)$ are non-zero linear delay-differential polynomials, $\min\{n,m\}\ge 2$, $a_1,a_2$ are non-zero small functions with relative to $f$ and $g$, or to $f^n(z)L(z,g)$ and $g^m(z)L(z,f)$, respectively. These results give answers to three open questions raised by Gao, Liu[Bull. Korean Math. Soc. 59 (2022)] and Liu, Liu[J. Math. Anal. Appl. 543 (2025)]. |
| title | Some new findings concerning value distribution of a pair of delay-differential polynomials |
| topic | Complex Variables 39A05, 30D35 |
| url | https://arxiv.org/abs/2504.06825 |