Notes on Generalized Grötzsch Ring Function and Generalized Hersch-Pfluger Distortion Function
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929739722129408 |
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| author | Bao, Qi Wang, MiaoKun |
| author_facet | Bao, Qi Wang, MiaoKun |
| contents | For $a\in(0,1)$, $r\in(0,1)$ and $K\in(1,\infty)$, let $μ_{a}(r)$ and $φ_{K}^{a}(r)$ be the generalized Grötzsch ring function and generalized Hersch-Pfluger distortion function. In the past few years, the functions $μ_{a}(r)$ and $φ_{K}^{a}(r)$, and their special cases $μ_{1/2}(r)$ and $φ_{K}^{1/2}(r)$ have been playing the very important role on the theory of quasiconformal mappings and (generalized) Ramanujan's modular equations. In this paper, we present a series expansion of $μ_{a}(r)$, and thus prove that the function $r\mapsto -[μ_{a}(r)-\log{(e^{R(a)/2})/r}]$ is absolutely monotonic on $(0,1)$. Here $R(a)$ is the Ramanujan constant. In addition, we also investigate the submultiplicative and power submultiplicative properties of $φ_{K}^{a}(r)$, and establish some new inequalities for $φ_{K}^{a}(r)$ in terms of elementary functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2202_09758 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Notes on Generalized Grötzsch Ring Function and Generalized Hersch-Pfluger Distortion Function Bao, Qi Wang, MiaoKun Classical Analysis and ODEs 11F03, 33C05 For $a\in(0,1)$, $r\in(0,1)$ and $K\in(1,\infty)$, let $μ_{a}(r)$ and $φ_{K}^{a}(r)$ be the generalized Grötzsch ring function and generalized Hersch-Pfluger distortion function. In the past few years, the functions $μ_{a}(r)$ and $φ_{K}^{a}(r)$, and their special cases $μ_{1/2}(r)$ and $φ_{K}^{1/2}(r)$ have been playing the very important role on the theory of quasiconformal mappings and (generalized) Ramanujan's modular equations. In this paper, we present a series expansion of $μ_{a}(r)$, and thus prove that the function $r\mapsto -[μ_{a}(r)-\log{(e^{R(a)/2})/r}]$ is absolutely monotonic on $(0,1)$. Here $R(a)$ is the Ramanujan constant. In addition, we also investigate the submultiplicative and power submultiplicative properties of $φ_{K}^{a}(r)$, and establish some new inequalities for $φ_{K}^{a}(r)$ in terms of elementary functions. |
| title | Notes on Generalized Grötzsch Ring Function and Generalized Hersch-Pfluger Distortion Function |
| topic | Classical Analysis and ODEs 11F03, 33C05 |
| url | https://arxiv.org/abs/2202.09758 |